Dynamics of sets of zero density - IMPA

Yuri Gomes LimaDynamics of sets of zero densityTese submetida ao Instituto Nacional deMatemática Pura e Aplicada - IMPA - paraobtenção do título de Doutor em Ciências.Orientador: Prof. Carlos Gustavo Moreira.Orientador: Prof. Enrique Ramiro Pujals.Rio de JaneiroMarço de 2011

To my beloved mom, Glaucia.

AcknowledgementsMy Ph.D. degree is part of at least ten years of my life, so I will acknowledge people from thevery ancient times.My academic career began in high school through mathematical olympiads, whose first encouragementscame from my 9th grade teacher Max and developments by Antonio Caminhaand Onofre Campos. Its an honor having them, nowadays, as true friends.Later on, I was admitted to the undergraduate studies at Universidade Federal do Ceará. Ithank Professor Abdênago Barros, who provided all the administrative tools necessary for myexchange from Engineering to Mathematics. During this time, I had the help of many professors,of which I highlight Plácido Andrade, João Lucas Barbosa, Abdênago Barros, AntonioCaminha, LuquésioJorge, Levi Lima, FábioMontenegro, Francisco Pimentel andRomildoSilva.In 2007, I began my graduate studies at IMPA, supported by one of my two advisors, CarlosGustavo Moreira - Gugu - and Marcelo Viana. Some months later I met my other advisor,Enrique Pujals. A huge acknowledgement is devoted to them, for their valuable mathematicalsupport, covering of conference expenses and advices in life. Yet more special thanks are givento my advisors, who (Lebesgue almost-every occasion) patiently heard my doubts and explainedin the simplest way; who gave me insights to the development of this thesis. I have learned alot from them, from the happiness of Gugu to the optimism of Enrique; from the social ideas ofGugu to the political ones of Enrique. Their flaws are that Enrique does not play football andGugu does terribly bad.In the middle of my stay at IMPA I had the opportunity of doing a semester internship atThe Ohio State University, under the supervision of Vitaly Bergelson. I would like to thankhim for such guidance, enormous encouragement and also to the suggestion of research topic,which result is Chapter 5 of the present work. I made many friends in Columbus, of which Ihighlight Naomi Adaniya, Fabian Benitez, Darcey Hull, Kathrin Levine, Daniel Munoz, SanjnaShah, Sylvia Silveira and Laura Tompkins.Carlos Matheus has been an alma mater to me. He teaches me a lot, specially in topics Idon’t have any knowledge. In addition, he shared his blog for posts about my research. He havehad many discussions, together with Aline Cerqueira - hearing everything, telling stories about“the dark side of Rio” and supporting us with her special care. I thank Alejandro Kocsard, forii

his patience and availability in my latest project and Jacob Palis for his influence and globalperspectives.I have many considerations to IMPA employees. As they are too many, I represent each departmentby one person: Guilherme Brondi, Nelly Carvajal, Fernando Codá, Alexandre Maria,Sonia Melo, Samantha Nunes, Ana Paula Rodrigues, Fatima Russo and Fábio Santos. Mathematicalfriends also contributed (a lot!) to the fulfillment of my Ph.D.: Alan Prata, Martin deBorbon, Fernando Carneiro, Thiago Catalan, Pablo Dávalos, Jorge Erick, Tertuliano Franco,Pablo Guarino, Marcelo Hilário, Cristina Lizana, Ives Macedo, José Manuel, Fernando Marek,Jyrko Morris, Adriana Neumann, Ivaldo Nunes, Carlo Pietro, Artem Raibekas, José Régis,Patrícia Romano, Waliston Silva, Wanderson Silva, Vanessa Simões, Ricardo Turolla and FranciscoValenzuela. I thank the financial support provided by Faperj-Brazil, IMPA and Pronex.To those who shared the apartment with me: Alex Abreu, Samuel Barbosa, Thiago Barros,Renan Finder and Helio Macedo. Yes, it demands a great effort to stand me. Thanks to themfor this and the allowance of transforming the apartment into “home, sweet home”.Outside the mathematical scenario, I would like to acknowledge my friends Bruno Cals,Emanuel Carneiro, Filipe Carlos, Eliza Dias, Sofia Galvão, Lívia Leal, Caio Leão, ClarissaLima, Renan Lima, João Marcelo, Davi Maximo, Mayara Mota, Rodrigo Nobre, José Ricardo,André Silveira, Cícero Thiago and Chico Vieira.The golden key of my acknowledgements go to the basis of everything, my family: parentsGlaucia, Raul and Gilberto; siblings Gisele and Patrick; aunts Fatinha, Gardênia; cousinsGabriela Gomes, Diego Lacerda, Igor Marinho, Harley Teixeira, Mardey Teixeira and RebecaTeixeira. They give me the strength and support that allow me to face the everyday life difficultiesand good moments.iii

“Ora (direis) ouvir estrelas! CertoPerdeste o senso!” E eu vos direi, no entanto,Que, para ouvi-las, muita vez despertoE abro as janelas, pálido de espanto...E conversamos toda a noite, enquantoA via láctea, como um pálio aberto,Cintila. E, ao vir do sol, saudoso e em pranto,Inda as procuro pelo céu deserto.Direis agora: “Tresloucado amigo!Que conversas com elas? Que sentidoTem o que dizem, quando estão contigo?”E eu vos direi: “Amai para entendê-las!Pois só quem ama pode ter ouvidoCapaz de ouvir e de entender estrelas”.Via Láctea, de Olavo Bilac.

AbstractThis thesis is devoted to the study of sets of zero density, both continuous - inside R - anddiscrete - inside Z. Most of the arguments are combinatorial in nature.The interests are twofold. The first investigates arithmetic sums: given two subsets E,F ofR or Z, what can be said about E+λF for most of the parameters λ ∈ R? The continuous casedates back to Marstrand. We give an alternative combinatorial proof of his theorem for theparticular case of products of regular Cantor sets and, in the sequel, extend these techniques togive the proof of the general case.We also discuss Marstrand’s theorem in discrete spaces. More specifically, we proposeafractaldimension for subsets of the integers and establish a Marstrand type theorem in this context.The second interest is ergodic theoretical. It contains constructions of Z d -actions withprescribed topological and ergodic properties, such as total minimality, total ergodicity andtotal strict ergodicity. These examples prove that Bourgain’s polynomial pointwise ergodictheorem has not a topological version.v

ResumoO presente trabalho estuda conjuntos de densidade zero, tanto contínuos - subconjuntos deR - quanto discretos - subconjuntos de Z. Os argumentos são, em sua maioria, de naturezacombinatória.Os interesses são divididos em duas partes. O primeira investiga somas aritméticas: dadossubconjuntos E,F de R ou Z, o que se podeinferir a respeito doconjunto E+λF paraamaioriados parâmetros λ ∈ R? O caso contínuo tem origem nos trabalhos de Marstrand. Aqui, é dadauma prova alternativa desse teorema no caso em que o conjunto é produto de dois conjuntosde Cantor regulares e, posteriormente, as técnicas aplicadas são estendidas para a obtenção deuma prova do caso geral.Ainda na primeira parte, discutimos o Teorema de Marstrand em espaços discretos. Maisespecificamente, propomos uma dimensão fractal para subconjuntos de inteiros e provamos umresultado do tipo Marstrand nesse contexto. Uma grande quantidade de exemplos é discutida,além de contraexemplos para possíveis extensões do resultado.Osegundointeresseéergódico. ConstruímosZ d -ações compropriedadesergódicasetopológicasprescritas,comototalmenteminimal, totalmenteergódicoetotalmenteestritamenteergódico.Tais exemplos provam que não existe uma versão topológica para o teorema de Bourgain sobrea convergência pontual de médias ergódicas polinomiais.vi

List of Publications Z d -actions with prescribed topological and ergodic properties. Toappear in ErgodicTheory& Dynamical Systems. A combinatorial proof of Marstrand’s Theorem for products of regular Cantor sets, withCarlos Gustavo Moreira. To appear in Expositiones Mathematicae. Yet another proof of Marstrand’s theorem, with Carlos Gustavo Moreira. To appear inBulletin of the Brazilian Mathematical Society. A Marstrand theorem for subsets of integers, with Carlos Gustavo Moreira. Available atwww.impa.br/∼yurilima.vii

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Contents1 Brief description 11.1 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Largeness in R and Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Zero density in R and Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 A combinatorial proof of Marstrand’s Theorem for products of regular Cantorsets 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Regular Cantor sets of class C 1+α . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Proof of Theorem 2.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Proof of Theorem 2.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Yet another proof of Marstrand’s Theorem 213.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.2 Dyadic squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.1 Rewriting the integral I i . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.2 Transversality condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.3 Good covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4 Proof of Theorems 3.1.1 and 3.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32ix

4 A Marstrand theorem for subsets of integers 334.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2.1 General notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2.2 Counting dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3.1 Polynomial subsets of Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3.2 Cantor sets in Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3.3 Generalized IP-sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4 Regularity and compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4.1 Regular sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4.2 Compatible sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4.3 A counterexample of Theorem 4.1.1 for regular non-compatible sets . . . 474.4.4 Another counterexample of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . 494.5 Proofs of the Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 Z d -actions with prescribed topological and ergodic properties 575.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2.1 Group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2.2 Subgroups of Z d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2.3 Symbolic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2.4 Topological entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2.5 Frequencies and unique ergodicity . . . . . . . . . . . . . . . . . . . . . . 655.2.6 Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.3 Main Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.3.1 Minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.3.2 Total minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.3.3 Total strict ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3.4 Proof of Theorem 5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.4 Proof of Theorems 5.1.2 and 5.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 765.4.1 Proof of Theorem 5.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.4.2 Proof of Theorem 5.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Chapter 1Brief descriptionThis thesis is devoted to the study of sets of zero density, both continuous - inside R - anddiscrete - inside Z. Most of the arguments are combinatorial in nature.The interests are twofold. The first investigates arithmetic sums: given two subsets E,F ofR or Z, what can be said about E+λF for most of the parameters λ ∈ R? The continuous casedates back to Marstrand, as we will describe in the next section. The discrete case is new [24].The second interest is ergodic theoretical. Specifically, it is shown that Bourgain’s polynomialpointwise ergodic theorem has not a topological version.Each chapter originated an article and for this reason is self-contained. Chapters 2, 3, 4 werewritten together with one of my Ph.D. advisors, Carlos Gustavo Moreira (Gugu), and Chapter5 by myself as I was a visiting scholar at The Ohio State University under the supervision ofVitaly Bergelson.The present chapter describes the main goals of this work and gives a brief description ofeach of the next chapters.1.1 GoalsConsider the following table.Continuous DiscretePositive density ̌ ̌Zero density ̌ ?For each box with the checkmark ̌, we represent a situation for which there is an activeresearch and a solid theory has been developed, as we will describe in the sequel. The only1

2unchecked box concerns discrete sets of zero density. These objects are of great importance, notonly for intrisic reasons, but also because many of them are natural arithmetic sets that appearin diverse areas of Mathematics, from which we mention• the prime numbers: even having zero density, B. Green and T. Tao [16] proved that the primenumbers are combinatorially rich in the sense that they contain infinitely long arithmeticprogressions;• J. Bourgain [6] has proved almost sure pointwise convergence of ergodic averages along polynomials.The above results strongly rely on the particular combinatorial and analytical structure ofeach set. In this thesis, we obtain results in a more general point of view. Yet concerning thetable above, let us contrast the continuous and discrete situations.1.1.1 Largeness in R and ZLet m denote the Lebesgue measure of R. We ask the following question: if K ⊂ R is large inthe sense of m, does it inherit some structure from R? A satisfactory answer is given byTheorem 1.1.1 (Lebesgue differentiation). If K ⊂ R is a Borel set with m(K) > 0, thenfor m-almost every x ∈ K.m(K ∩(x−ε,x+ε))lim= 1ε→0 + 2εIn other words, copies of intervals of the real line “almost” appear in large sets of R. Surprisingly,the same phenomenon holds for subsets of Z. Given a subset E ⊂ Z, let d ∗ (E) denotethe upper-Banach density of E, defined asd ∗ |E ∩{M +1,...,N}|(E) = limsup·N−M→∞ N −MWe thus have a discrete version of Theorem 1.1.1.Theorem 1.1.2 (Szemerédi [45]). If E ⊂ Z has positive upper Banach density, then E containsarbitrarily long arithmetic progressions.One can interpret this result by saying that density represents the correct notion of largenessneeded to preserve finite configurations of Z. These two situations just described are examplesof Ramsey’s principle which, roughly speaking, asserts thatIf a substructure occupies a positive portion of a structure, then it must contain “copies” ofthe structure.

31.1.2 Zero density in R and ZHowever, theorems 1.1.1 and 1.1.2 can not infer anything if the density is zero, but these setsmight posses rich combinatorial properties as well. For example, the ternary Cantor set⎧⎫⎨∑⎬K = a⎩ i ·3 −i ; a i = 0 or 2⎭n≥1satisfies K + K = [0,2] and naturally appears in the theory of homoclonic bifurcations onsurfaces. Regarding the continuous case, one can investigate largeness of a set of zero densityvia its Hausdorff dimension. It is a classical result in Geometric Measure Theory the followingTheorem 1.1.3 (Marstrand [27]). If K ⊂ R 2 is a Borel set with Hausdorff dimension greaterthan one, then the projection of K into R in the direction of λ has positive Lebesgue measurefor m-almost every λ ∈ R.Informally speaking, this means that sets of Hausdorff dimension greater than one are “fat”in almost every direction. In particular, if K = K 1 × K 2 is a cartesian product of two onedimensionalsubsets of R, Marstrand’s theorem translates to “m(K 1 +λK 2 ) > 0 for m-almostevery λ ∈ R”.Up to now, a theory of fractal sets in Z has not been developed. This constitutes the maingoal of this thesis. We will investigate these sets from geometrical and ergodic points of view,according to the diagram below.PART 1: GEOMETRYMarstrand’s theorem for subsets of Z↑Marstrand’s theorem in R: new proofPART 2: ERGODICErgodic theorems along sets of zero density↑Z d -actions with prescribed topological and ergodic properties

4Part 1 is the content of chapters 2, 3 and 4 and Part 2 of chapter 5. In order to obtain aMarstrand’s theorem for subsets of Z, we first establish a new proof of the classical Marstrand’stheorem that can be adapted to the discrete case. Regarding Part 2, we prove that Bourgain’spolynomial pointwise ergodic theorem has not a topological version by constructing Z d -actionswith prescribed tolopogical and ergodic properties. In the next sections of this chapter, we willgive a brief description of the ideas involved.1.2 Chapter 2Let m denote the Lebesgue measure of R. It is a classical result in Geometric Measure Theorythe followingTheorem 1.2.1 (Marstrand [27]). If K ⊂ R 2 is a Borel set with Hausdorff dimension greaterthan one, then the projection of K into R in the direction of λ has positive Lebesgue measurefor m-almost every λ ∈ R.In particular, if K = K 1 × K 2 is a cartesian product of two one-dimensional subsets ofR, Marstrand’s theorem translates to “m(K 1 + λK 2 ) > 0 for m-almost every λ ∈ R”. Theinvestigation of such arithmetic sums K 1 + λK 2 has been an active area of Mathematics, inspecial when K 1 and K 2 are dynamically defined Cantor sets. Remarkably, M. Hall [18] proved,in 1947, that the Lagrange spectrum 1 contains a whole half line, by showing that the arithmeticsum K(4)+K(4) of a certain Cantor set K(4) ⊂ R with itself contains [6,∞).Marstrand’s Theorem for product of Cantor sets is also fundamental in certain results ofdynamical bifurcations, namely homoclinic bifurcations in surfaces. For instance, in [38] it isused to prove that hyperbolicity holds for most small positive values of parameter in homoclinicbifurcations associated to horseshoes with Hausdorff dimension smaller than one; in [39] itis used to show that hyperbolicity is not prevalent in homoclinic bifurcations associated tohorseshoes with Hausdorff dimension larger than one.Looking for some kind of converse to Marstrand’s theorem, J. Palis conjectured in [36] thatfor generic pairs of regular Cantor sets (K 1 ,K 2 ) of the real line either K 1 +K 2 has zero measureor else it contains an interval. Observe that the first part is a consequence of Marstrand’stheorem. In [33], it is proved that stable intersections of regular Cantor sets are dense in theregion where the sum of their Hausdorff dimensions is larger than one, thus establishing theremaining of Palis conjecture. It is remarkable to point out that Marstrand’s theorem is usedin the proof.1 The Lagrange spectrum is the set of best constants of rational approximations of irrational numbers. See [9]for the specific description.

5In the connection of these two applications, I would like to point out that a formula for theHausdorff dimension of K 1 +K 2 , under mild assumptions of non-linear Cantor sets K 1 and K 2 ,has been obtained by Gugu [31] and applied in [32] to prove that the Hausdorff dimension ofthe Lagrange spectrum increases continuously. In parallel to this non-linear setup, Y. Peres andP. Shmerkin proved the same phenomena happen to self-similar Cantor sets without algebraicresonance [41]. Finally, M. Hochman and P. Shmerkin extended and unified many resultsconcerning projections of products of self-similar measures on regular Cantor sets [19].Thetechniques used in[31] werebased onestimates obtained in [33]. With these tools, Guguand myself were able to develop in [25] a purely combinatorial proof of Marstrand’s theorem forproducts of regular dynamically defined Cantor sets. Denoting by HD the Hausdorff dimension,the main theorem isTheorem 1.2.2 (Lima-Moreira [25]). If K 1 ,K 2 are regular dynamically defined Cantor sets ofclass C 1+α , α > 0, such that d = HD(K 1 )+HD(K 2 ) > 1, then m(K 1 +λK 2 ) > 0 for m-almostevery λ ∈ R.Let m d denote the Hausdorff d-measure on R. The idea of the proof relies on the fact thata regular Cantor set of Hausdorff dimension d is regular in the sense that the m d -measure ofarbitrarily small portions of it can be uniformly controlled. Once this is true, a double countingargument proves that the arithmetic sum has Lebesgue measure bounded away from zero atadvanced stages of the construction of the Cantor sets, except for a small set of parameters.The good feature of the proof is that this discretization idea may be applied to the discretecontext, as we shall see in Chapter 4.The same idea of homogeneity is applied in [26] to prove the general case of Marstrand’stheorem. Namely, once some upper regularity on the m d -measure of K is assumed, it is possibleto apply a weighted version of [25] (the covers of K may be composed of subsets with differentdiameter scales) to conclude the result.1.3 Chapter 3Chapter 3 extends the techniques developed in Chapter 2 and encloses a combinatorial proofof Marstrand’s Theorem without any restrictions on K. The proof makes a study on the onedimensionalfibers of K in every direction and relies on two facts:(I) Transversality condition: given two squares on the plane, the Lebesgue measure of the setof angles θ ∈ R for which their projections in the direction of θ have nonempty intersection hasan upper bound.(II) After a regularization of K, (I) enables us to conclude that, except for a small set of anglesθ ∈ R, the fibers in the direction of θ are not concentrated in a thin region. As a consequence,

6K projects into a set of positive Lebesgue measure.The idea of (II) is based on [31] and was employed in [26] to develop a combinatorial proofof Marstrand’s Theorem when K is the product of two regular Cantor sets (which is the contentof Chapter 2).Compared to other proofs of Marstrand’s Theorem, the new ideas here are the discretizationof the argument and the use of dyadic covers, which allow the simplification of the methodemployed.1.4 Chapter 4Given a subset E ⊂ Z, let d ∗ (E) denote the upper-Banach density of E, defined asd ∗ |E ∩{M +1,...,N}|(E) = limsup·N−M→∞ N −MSzemerédi’s theorem asserts that if d ∗ (E) > 0, then E contains arbitrarily long arithmeticprogressions. Onecan interpretthisresultbysayingthat densityrepresentsthecorrect notion oflargeness needed to preserve finite configurations of Z. On the other hand, Szemerédi’s theoremcannot infer any property about subsets of zero density, and many of these are important.A class of examples are the integer polynomial sequences. The prime numbers are anotherexample. These sets may, as well, contain combinatorially rich patterns.A set E ⊂ Z of zero upper-Banach density is characterized as occupying portions in intervalsof Z that grow in a sublinear fashion as the length of the intervals grow. On the other hand,there still may exist some kind of growth speed. For example, the number of perfect squareson an interval of length n is about n 0.5 . This exponent means, in some sense, a dimension of{n 2 ; n ∈ Z} inside Z. In general, define the counting dimension of E ⊂ Z asD(E) = limsup|I|→∞log|E ∩I|log|I|,where I runs over the intervals of Z.In [24], Gugu and myself investigated this dimension when one considers arithmetic sumsof the form E +⌊λF⌋, λ ∈ R, for a class of subsets E,F ⊂ Z. At first sight, by an ingenuousapplication of the multiplicative principle, one should expect that E +⌊λF⌋ has dimension atleast D(E)+D(F). This is not true at all, due to two reasons:(1) E and F may be very irregular as subsets of Z.(2) E and F may exhibit growth speed inside intervals of very different lengths.

7This led to the definition of regular compatible sets: E is regular if|E ∩I| 1,|I|D(E)where I runs over all intervals of Z, and there exists a sequence of intervals (I n ) n≥1 such that|I n | → ∞ and|E ∩I n | 1. (1.4.1)|I n | D(E)Two regular subsets E,F ⊂ Z are compatible if there exist two sequences (I n ) n≥1 , (J n ) n≥1of intervals with increasing lengths satisfying (1.4.1) and such that I n ∼ J n (above, we usedasymptotic Vinogradov notations and ∼). The main theorem of [24] isTheorem 1.4.1 (Lima-Moreira [24]). Let E,F ⊂ Z be two regular compatible sets. ThenD(E +⌊λF⌋) ≥ min{1,D(E) +D(F)}for Lebesgue almost every λ ∈ R. If in addition D(E)+D(F) > 1, then E +⌊λF⌋ has positiveupper-Banach density for Lebesgue almost every λ ∈ R.The above theorem has direct consequences when applied to integer polynomial sequences.For any p(x) ∈ Z[x] of degree d, the set E = {p(n); n ∈ Z} is regular and has dimension 1/d.Also, it is compatible with any other regular set. These observations imply the second result ofthe chapter.Theorem 1.4.2 (Lima-Moreira [24]). Let p i ∈ Z[x] with degree d i and let E i = {p i (n); n ∈ Z},i = 0,...,k. If1d 0+ 1 d 1+···+ 1 d k> 1,then the arithmetic sum E 0 + ⌊λ 1 E 1 ⌋ + ··· + ⌊λ k E k ⌋ has positive upper-Banach density forLebesgue almost every (λ 1 ,...,λ k ) ∈ R k .1.5 Chapter 5LetT beamesure-preservingtransformationon theprobability space(X,B,µ) andf : X → Rameasurable function. A successful area in ergodic theory deals with the convergence of averagesn −1 ·∑nk=1 f ( T k x ) , x ∈ X, when n converges to infinity. The well known Birkhoff’s Theoremstates that such limit exists for almost every x ∈ X whenever f is an L 1 -function. Severalresults have been (and still are being) proved when, instead of {1,2,...,n}, average is madealong other sequences of natural numbers. A remarkable result on this direction was given by J.Bourgain [6], where he proved that if p(x) is a polynomial with integer coefficients and f is an

L p -function, for some p > 1, then the averages n −1·∑nk=1 f ( T p(k) x ) converge for almost everyx ∈ X. In other words, convergence fails to hold for a negligible set with respect to the measureµ. We mention this result is not true for L 1 -functions, as recently proved by Z. Buczolich andD. Mauldin [7].In [3], V. Bergelson asked if this set is also negligible from the topological point of view.It turned out, by a result of R. Pavlov [40], that this is not true. He proved that, for everysequence (p n ) n≥1 ⊂ Z of zero upper-Banach density, there exist a totally minimal, totallyuniquely ergodic and topologically mixing transformation (X,T) and a continuous functionf : X → R such that n −1 · ∑nk=1 f (Tp kx) fails to converge for a residual set of x ∈ X. Theconditions of minimality, unique ergodicity, although they seem artificial, are natural for thequestion posed by Bergelson, in order to avoid pathological counterexamples. The constructionof Pavlov was inspired in results of Hahn and Katznelson [17].In [23], I extended the results of Pavlov [40] to the setting of Z d -actions. The methodconsisted of constructing (totally) minimal and (totally) uniquely ergodic Z d -actions. Theprogram was carried out by constructing closed shift invariant subsets of a sequence space.More specifically, a sequence of finite configurations (C k ) k≥1 of {0,1} Zd was built, where C k+1is essentially formed by the concatenation of elements in C k such that each of them occursstatistically well-behaved in each element of C k+1 . Finally, the set of limits of shifted C k -configurations as k → ∞ constitutes the shift invariant subset. The main results on [23] isTheorem 1.5.1 (Lima [23]). Given a set P ⊂ Z d of zero upper-Banach density, there exist atotally strictly ergodic Z d -action (X,A,µ,T) and a continuous function f : X → R such thatthe ergodic averages1|P ∩(−n,n) d |fail to converge for a residual set of x ∈ X.∑g∈P∩(−n,n) d f (T g x)Yet in the arithmetic setup, let p 1 ,...,p d ∈ Z[x]. Bergelson and Leibman proved in[4] that if T is a minimal Z d -action, then there is a residual set Y ⊂ X for which x ∈{T (p 1(n),...,p d (n)) x; n ∈ Z}, for every x ∈ Y. The same method employed in Theorem 1.5.1also gives the best topological result one can expect in this context.Theorem 1.5.2 (Lima [23]). Given a set P ⊂ Z d of zero upper-Banach density, there existsa totally strictly ergodic Z d -action (X,A,µ,T) and an uncountable set Y ⊂ X for which x ∉{T p x; p ∈ P}, for every x ∈ Y.8

Chapter 2A combinatorial proof ofMarstrand’s Theorem for productsof regular Cantor setsIn a paper from 1954 Marstrand proved that if K ⊂ R 2 has Hausdorff dimension greater than1, then its one-dimensional projection has positive Lebesgue measure for almost-all directions.In this chapter, we give a combinatorial proof of this theorem when K is the product of regularCantor sets of class C 1+α , α > 0, for which the sum of their Hausdorff dimension is greaterthan 1.2.1 IntroductionIf U is a subset of R n , the diameter of U is |U| = sup{|x−y|;x,y ∈ U} and, if U is a family ofsubsets of R n , the diameter of U is defined as‖U‖ = sup |U|.U∈UGiven d > 0, the Hausdorff d-measure of a set K ⊆ R n ism d (K) = lim(∑infε→0 U covers K‖U‖ d 0 . We define the Hausdorff dimension of K as HD(K) = d 0 .Also, for each θ ∈ R, let v θ = (cosθ,sinθ), L θ the line in R 2 through the origin containing v θ9.

Chapter 2. Marstrand’s Theorem for products of Cantor sets 10and proj θ : R 2 → L θ the orthogonal projection. From now on, we’ll restrict θ to the interval[−π/2,π/2], because L θ = L θ+π .In 1954, J. M. Marstrand [27] proved the following result on the fractal dimension of planesets.Theorem. If K ⊆ R 2 is a Borel set such that HD(K) > 1, then m(proj θ (K)) > 0 for m-almostevery θ ∈ R.The proof is based on a qualitative characterization of the “bad” angles θ for which theresult is not true. Specifically, Marstrand exhibits a Borel measurable function f(x,θ), (x,θ) ∈R 2 ×[−π/2,π/2], such that f(x,θ) = ∞ for m d -almost every x ∈ K, for every “bad” angle. Inparticular,∫f(x,θ)dm d (x) = ∞. (2.1.1)On the other hand, using a version of Fubini’s Theorem, he proves that∫ π/2 ∫dθ f(x,θ)dm d (x) = 0K−π/2Kwhich, in view of (2.1.1), implies thatm({θ ∈ [−π/2,π/2]; m(proj θ (K)) = 0}) = 0.These results are based on the analysis of rectangular densities of points.Many generalizations and simpler proofs have appeared since. One of them appeared in1968 by R. Kaufman who gave a very short proof of Marstrand’s theorem using methods ofpotential theory. See [21] for his original proof and [29], [37] for further discussion.In this chapter, we prove a particular case of Marstrand’s Theorem.Theorem 2.1.1. If K 1 ,K 2 are regular Cantor sets of class C 1+α , α > 0, such that d =HD(K 1 )+HD(K 2 ) > 1, then m(proj θ (K 1 ×K 2 )) > 0 for m-almost every θ ∈ R.The argument also works to show that the push-forward measure of the restriction of m d toK 1 ×K 2 , defined as µ θ = (proj θ ) ∗ (m d | K1 ×K 2), is absolutely continuous with respect to m, form-almost every θ ∈ R. Denoting its Radon-Nykodim derivative by χ θ = dµ θ /dm, we also giveyet another proof of the following result.Theorem 2.1.2. χ θ is an L 2 function for m-almost every θ ∈ R.Our proof makes a study on the fibers proj −1 θ (v)∩(K 1 ×K 2 ), (θ,v) ∈ R×L θ , and relieson two facts:(I) A regular Cantor set of Hausdorff dimension d is regular in the sense that the m d -measureof small portions of it has the same exponential behavior.

Chapter 2. Marstrand’s Theorem for products of Cantor sets 11(II) This enables us to conclude that, except for a small set of angles θ ∈ R, the fibersproj θ −1 (v) ∩ (K 1 × K 2 ) are not concentrated in a thin region. As a consequence, K 1 × K 2projects into a set of positive Lebesgue measure.The idea of (II) is based on [31]. It proves that, if K 1 and K 2 are regular Cantor sets of classC 1+α , α > 0, and at least one of them is non-essentially affine (a technical condition), then thearithmetic sum K 1 +K 2 = {x 1 +x 2 ;x 1 ∈ K 1 ,x 2 ∈ K 2 } has the expected Hausdorff dimension:HD(K 1 +K 2 ) = min{1,HD(K 1 )+HD(K 2 )}.Marstrand’s Theorem for products of Cantor sets is a source of very relevant applicationsin dynamical systems, as pointed out in Chapter 1. It is of fundamental importance in certainresults of dynamical bifurcations, namely homoclinic bifurcations. For instance, in [38] it isused to prove that hyperbolicity holds for most small positive values of parameter in homoclinicbifurcationsassociatedtohorseshoeswithHausdorffdimensionsmallerthanone; in[39]itisusedto show that hyperbolicity is not prevalent in homoclinic bifurcations associated to horseshoeswith Hausdorff dimension larger than one; in [33] it is used to prove that stable intersectionsof regular Cantor sets are dense in the region where the sum of their Hausdorff dimensions islarger than one; in [34] to show that, for homoclinic bifurcations associated to horseshoes withHausdorff dimension larger than one, typically there are open sets of parameters with positiveLebesgue density at the initial bifurcation parameter corresponding to persistent homoclinictangencies.2.2 Regular Cantor sets of class C 1+αWe say that K ⊂ R is a regular Cantor set of class C 1+α , α > 0, if:(i) there are disjoint compact intervals I 1 ,I 2 ,...,I r ⊆ [0,1] such that K ⊂ I 1 ∪···∪I r andthe boundary of each I i is contained in K;(ii) there is a C 1+α expanding map ψ defined in a neighbourhood of I 1 ∪ I 2 ∪ ··· ∪ I r suchthat ψ(I i ) is the convex hull of a finite union of some intervals I j , satisfying:by(ii.1) for each i ∈ {1,2,...,r} and n sufficiently big, ψ n (K ∩I i ) = K;(ii.2) K = ⋂ ψ −n (I 1 ∪I 2 ∪···∪I r ).n∈NThe set {I 1 ,...,I r } is called a Markov partition of K. It defines an r ×r matrix B = (b ij )b ij = 1, if ψ(I i ) ⊇ I j= 0, if ψ(I i )∩I j = ∅,

Chapter 2. Marstrand’s Theorem for products of Cantor sets 12which encodes the combinatorial properties of K. Given such matrix, consider the set Σ B ={θ = (θ1 ,θ 2 ,...) ∈ {1,...,r} N ; b θi θ i+1= 1,∀i ≥ 1 } and the shift transformation σ : Σ B → Σ Bgiven by σ(θ 1 ,θ 2 ,...) = (θ 2 ,θ 3 ,...).There is a natural homeomorphism between the pairs (K,ψ) and (Σ B ,σ). For each finiteword a = (a 1 ,...,a n ) such that b ai a i+1= 1, i = 1,...,n−1, the intersectionI a = I a1 ∩ψ −1 (I a2 )∩···∩ψ −(n−1) (I an )is a non-empty interval with diameter |I a | = |I an |/|(ψ n−1 ) ′ (x)| for some x ∈ I a , which isexponentially small if n is large. Then, {h(θ)} = ⋂ n≥1 I (θ 1 ,...,θ n) defines a homeomorphismh : Σ B → K that commutes the diagramΣ Bσ Σ Bh h K ψKIf λ = sup{|ψ ′ (x)|;x ∈ I 1 ∪··· ∪I r } ∈ (1,∞), then ∣ ∣I (θ1∣,...,θ n+1 ) ≥ λ ·∣∣I −1 (θ1∣,...,θ n) and so, forρ > 0 small and θ ∈ Σ B , there is a positive integer n = n(ρ,θ) such thatρ ≤ ∣ I(θ1 ∣,...,θ n) ≤ λρ.Definition 2.2.1. A ρ-decomposition of K is any finite set (K) ρ = {I 1 ,I 2 ,...,I r } of disjointclosed intervals of R, each one of them intersecting K, whose union covers K and such thatρ ≤ |I i | ≤ λρ, i = 1,2,...,r.Remark 2.2.2. Although ρ-decompositions are not unique, we use, for simplicity, the notation(K) ρ todenoteanyofthem. Wealsousethesamenotation(K) ρ todenotetheset ⋃ I∈(K) ρI ⊂ Rand the distinction between these two situations will be clear throughout the text.Every regular Cantor set{of class C 1+α } has a ρ-decomposition for ρ > 0 small: by thecompactness of K, the family I (θ1 ,...,θ n(ρ,θ))has a finite cover (in fact, it is only necessaryθ∈Σ Bfor ψ to be of class C 1 ). Also, one can define ρ-decomposition for the product of two Cantorsets K 1 and K 2 , denoted by (K 1 × K 2 ) ρ . Given ρ ≠ ρ ′ and two decompositions (K 1 × K 2 ) ρ ′and (K 1 ×K 2 ) ρ , consider the partial order(K 1 ×K 2 ) ρ ′ ≺ (K 1 ×K 2 ) ρ ⇐⇒ ρ ′ < ρ and⋃Q ′ ∈(K 1 ×K 2 ) ρ ′In this case, proj θ ((K 1 ×K 2 ) ρ ′) ⊆ proj θ ((K 1 ×K 2 ) ρ ) for any θ.Q ′ ⊆⋃Q∈(K 1 ×K 2 ) ρQ.A remarkable property of regular Cantor sets of class C 1+α , α > 0, is bounded distortion.

Chapter 2. Marstrand’s Theorem for products of Cantor sets 13Lemma 2.2.3. Let (K,ψ) be a regular Cantor set of class C 1+α , α > 0, and {I 1 ,...,I r } aMarkov partition. Given δ > 0, there exists a constant C(δ) > 0, decreasing on δ, with thefollowing property: if x,y ∈ K satisfy(i) |ψ n (x)−ψ n (y)| < δ;(ii) The interval [ψ i (x),ψ i (y)] is contained in I 1 ∪···∪I r , for i = 0,...,n−1,thenIn addition, C(δ) → 0 as δ → 0.e −C(δ) ≤ |(ψn ) ′ (x)||(ψ n ) ′ (y)| ≤ eC(δ) .A direct consequence of bounded distortion is the required regularity of K, contained in thenext result.Lemma 2.2.4. Let K be a regular Cantor set of class C 1+α , α > 0, and let d = HD(K). Then0 < m d (K) < ∞. Moreover, there is c > 0 such that, for any x ∈ K and 0 ≤ r ≤ 1,c −1 ·r d ≤ m d (K ∩B r (x)) ≤ c·r d .ThesamehappensforproductsK 1 ×K 2 of Cantor sets (without lossof generality, consideredwith the box norm).Lemma 2.2.5. Let K 1 ,K 2 be regular Cantor sets of class C 1+α , α > 0, and let d = HD(K 1 )+HD(K 2 ). Then 0 < m d (K 1 ×K 2 ) < ∞. Moreover, there is c 1 > 0 such that, for any x ∈ K 1 ×K 2and 0 ≤ r ≤ 1,c 1−1 ·r d ≤ m d ((K 1 ×K 2 )∩B r (x)) ≤ c 1 ·r d .See chapter 4 of [37] for the proofs of these lemmas. In particular, if Q ∈ (K 1 ×K 2 ) ρ , thereis x ∈ (K 1 ∪K 2 )∩Q such that B λ −1 ρ(x) ⊆ Q ⊆ B λρ (x) and so(c 1 λ d) −1·ρ d ≤ m d ((K 1 ×K 2 )∩Q) ≤ c 1 λ d ·ρ d .Changing c 1 by c 1 λ d , we may also assume thatc 1−1 ·ρ d ≤ m d ((K 1 ×K 2 )∩Q) ≤ c 1 ·ρ d ,which allows us to obtain estimates on the cardinality of ρ-decompositions.Lemma 2.2.6. Let K 1 ,K 2 be regular Cantor sets of class C 1+α , α > 0, and let d = HD(K 1 )+HD(K 2 ). Then there is c 2 > 0 such that, for any ρ-decomposition (K 1 × K 2 ) ρ , x ∈ K 1 × K 2and 0 ≤ r ≤ 1,( ) r#{Q ∈ (K 1 ×K 2 ) ρ ;Q ⊆ B r (x)} ≤ c 2 ·d·ρIn addition, c 2−1 ·ρ −d ≤ #(K 1 ×K 2 ) ρ ≤ c 2 ·ρ −d .

Chapter 2. Marstrand’s Theorem for products of Cantor sets 14Proof. We havec 1 ·r d ≥ m d ((K 1 ×K 2 )∩B r (x))∑≥ m d ((K 1 ×K 2 )∩Q)≥Q⊆B r(x)∑Q⊆B r(x)c 1−1 ·ρ d= #{Q ∈ (K 1 ×K 2 ) ρ ;Q ⊆ B r (x)}·c 1−1 ·ρ dand then( ) r#{Q ∈ (K 1 ×K 2 ) ρ ;Q ⊆ B r (x)} ≤ c 12 ·d·ρOn the other hand,m d (K 1 ×K 2 ) =∑∑m d ((K 1 ×K 2 )∩Q) ≤ c 1 ·ρ d ,Q∈(K 1 ×K 2 ) ρ Q∈(K 1 ×K 2 ) ρimplying that#(K 1 ×K 2 ) ρ ≥ c 1−1 ·m d (K 1 ×K 2 )·ρ −d .Taking c 2 = max{c 1 2 , c 1 /m d (K 1 ×K 2 )}, we conclude the proof.2.3 Proof of Theorem 2.1.1Given rectangles Q and ˜Q, let}Θ = Q,˜Q{θ ∈ [−π/2,π/2];proj θ (Q)∩proj θ (˜Q) ≠ ∅ .Lemma 2.3.1. If Q, ˜Q ∈ (K 1 ×K 2 ) ρ and x ∈ (K 1 ×K 2 )∩Q,˜x ∈ (K 1 ×K 2 )∩ ˜Q, then( )ρm Θ Q,˜Q≤ 2πλ·d(x,˜x) ·Proof. Consider the figure.xθ |θ −ϕ 0|˜xL θproj θ (˜x)θproj θ (x)

Chapter 2. Marstrand’s Theorem for products of Cantor sets 15Since proj θ (Q) has diameter at most λρ, d(proj θ (x),proj θ (˜x)) ≤ 2λρ and then, by elementarygeometry,sin(|θ −ϕ 0 |) = d(proj θ(x),proj θ (˜x))d(x,˜x)ρ≤ 2λ·d(x,˜x)ρ=⇒ |θ −ϕ 0 | ≤ πλ·d(x,˜x) ,because sin −1 y ≤ πy/2. As ϕ 0 is fixed, the lemma is proved.We point out that, although simple, Lemma 2.3.1 expresses the crucial property of transversalitythat makes the proof work, and all results related to Marstrand’s theorem use a similaridea in one way or another. See [43] where this tranversality condition is also exploited.Fixed a ρ-decomposition (K 1 ×K 2 ) ρ , let{N (K1 ×K 2 ) ρ(θ) = # (Q, ˜Q)}∈ (K 1 ×K 2 ) ρ ×(K 1 ×K 2 ) ρ ;proj θ (Q)∩proj θ (˜Q) ≠ ∅for each θ ∈ [−π/2,π/2] andE((K 1 ×K 2 ) ρ ) =∫ π/2−π/2N (K1 ×K 2 ) ρ(θ)dθ.Proposition 2.3.2. Let K 1 ,K 2 be regular Cantor sets of class C 1+α , α > 0, and let d =HD(K 1 )+HD(K 2 ). Then there is c 3 > 0 such that, for any ρ-decomposition (K 1 ×K 2 ) ρ ,E((K 1 ×K 2 ) ρ ) ≤ c 3 ·ρ 1−2d .Proof. Let s 0 = ⌈ log 2 ρ −1⌉ and choose, for each Q ∈ (K 1 ×K 2 ) ρ , a point x ∈ (K 1 ×K 2 )∩Q.By a double counting and using Lemmas 2.2.6 and 2.3.1, we have∑ ( )E((K 1 ×K 2 ) ρ ) = Θ Q,˜Q=Q,˜Q∈(K 1 ×K 2 ) ρms 0 ∑s=1∑mQ, ˜Q∈(K 1 ×K 2 )ρ2 −s 1, c 3 = 2 d+1 πλc 22 ·∑s≥1 2s(1−d) < +∞ satisfies the required inequality.

Chapter 2. Marstrand’s Theorem for products of Cantor sets 16This implies that, for each ε > 0, the upper boundN (K1 ×K 2 ) ρ(θ) ≤ c 3 ·ρ 1−2d(2.3.1)εholds for every θ except for a set of measure at most ε. Letting c 4 = c−2 2 ·c −1 3 , we will showthatm(proj θ ((K 1 ×K 2 ) ρ )) ≥ c 4 ·ε (2.3.2)for every θ satisfying (2.3.1). For this, divide [−2,2] ⊆ L θ in ⌊4/ρ⌋ intervals J ρ 1 ,..., Jρ ⌊4/ρ⌋ ofequal lenght (at least ρ) and defines ρ,i = #{Q ∈ (K 1 ×K 2 ) ρ ; proj θ (x) ∈ J ρ i}, i = 1,...,⌊4/ρ⌋.Then ∑ ⌊4/ρ⌋i=1s ρ,i = #(K 1 ×K 2 ) ρ and⌊4/ρ⌋∑i=1s ρ,i 2 ≤ N (K1 ×K 2 ) ρ(θ) ≤ c 3 ·ρ 1−2d ·ε −1 .Let S ρ = {1 ≤ i ≤ ⌊4/ρ⌋;s ρ,i > 0}. By Cauchy-Schwarz inequality,⎛ ⎞⎝ ∑ 2s ρ,i⎠i∈S ρ#S ρ ≥∑i∈S ρs ρ,i2≥ c 2 −2 ·ρ −2dc 3 ·ρ 1−2d ·ε −1 = c 4 ·ε·ρFor each i ∈ S ρ , the interval J ρ iis contained in proj θ ((K 1 ×K 2 ) ρ ) and thenwhich proves (2.3.2).m(proj θ ((K 1 ×K 2 ) ρ )) ≥ c 4 ·ε,Proof of Theorem 2.1.1. Fix a decreasing sequence(K 1 ×K 2 ) ρ1 ≻ (K 1 ×K 2 ) ρ2 ≻ ··· (2.3.3)of decompositions such that ρ n → 0 and, for each ε > 0, consider the sets{G n 1−2d ε = θ ∈ [−π/2,π/2]; N (K1 ×K 2 ) ρn(θ) ≤ c 3 ·ρ n ·ε −1} , n ≥ 1.Then m([−π/2,π/2]\G n ε) ≤ ε, and the same holds for the setG ε = ⋂ ∞⋃G l ε.n≥1If θ ∈ G ε , thenl=nm(proj θ ((K 1 ×K 2 ) ρn )) ≥ c 4 ·ε, for infinitely many n,which implies that m(proj θ (K 1 ×K 2 )) ≥ c 4 · ε. Finally, the set G = ⋃ n≥1 G 1/n satisfiesm([−π/2,π/2]\G) = 0 and m(proj θ (K 1 ×K 2 )) > 0, for any θ ∈ G.

Chapter 2. Marstrand’s Theorem for products of Cantor sets 172.4 Proof of Theorem 2.1.2Given any X ⊂ K 1 ×K 2 , let (X) ρ be the restriction of the ρ-decomposition (K 1 ×K 2 ) ρ to thoserectangles which intersect X. As done in Section 2.3, we’ll obtain estimates on the cardinalityof (X) ρ . Being a subset of K 1 × K 2 , the upper estimates from Lemma 2.2.6 also hold for X.The lower estimate is given byLemma 2.4.1. Let X be a subset of K 1 ×K 2 suchthat m d (X) > 0. Then there is c 6 = c 6 (X) > 0such that, for any ρ-decomposition (K 1 ×K 2 ) ρ and 0 ≤ r ≤ 1,c 6 ·ρ −d ≤ #(X) ρ ≤ c 2 ·ρ −d .Proof. As m d (X) < ∞, there exists c 5 = c 5 (X) > 0 (see Theorem 5.6 of [11]) such thatm d (X ∩B r (x)) ≤ c 5 ·r d , for all x ∈ X and 0 ≤ r ≤ 1,and thenm d (X) = ∑Q∈(X) ρm d (X ∩Q) ≤ ∑Q∈(X) ρc 5 ·(λρ) d =(c 5 ·λ d)·ρ d ·#(X) ρ .Just take c 6 = c−1 5 ·λ −d ·m d (X).Proposition 2.4.2. The measure µ θ = (proj θ ) ∗ (m d | K1 ×K 2) is absolutely continuous with respectto m, for m-almost every θ ∈ R.Proof. Note that the implicationX ⊂ K 1 ×K 2 , m d (X) > 0 =⇒ m(proj θ (X)) > 0 (2.4.1)is sufficient for the required absolute continuity. In fact, if Y ⊂ L θ satisfies m(Y) = 0, thenµ θ (Y) = m d (X) = 0,where X = proj −1 θ (Y). Otherwise, by (2.4.1) we would have m(Y) = m(proj θ (X)) > 0,contradicting the assumption.We prove that (2.4.1) holds for every θ ∈ G, where G is the set defined in the proof ofTheorem 2.1.1. The argument is the same made after Proposition 2.3.2: as, by the previouslemma, #(X) ρ has lower and upper estimates depending only on X and ρ, we obtain that−1 m(proj θ ((X) ρn )) ≥ c 3 ·c 62 ·ε, for infinitely many n,and then m(proj θ (X)) > 0.

Chapter 2. Marstrand’s Theorem for products of Cantor sets 18Let χ θ = dµ θ /dm. In principle, this is a L 1 function. We prove that it is a L 2 function, forevery θ ∈ G.Proof of Theorem 2. Let θ ∈ G 1/m , for some m ∈ N. ThenN (K1 ×K 2 ) ρn(θ) ≤ c 3 ·ρ n1−2d ·m, for infinitely many n. (2.4.2)For each of these n, consider the partition P n = {J ρn1 ,...,Jρn ⌊4/ρ n⌋ } of [−2,2] ⊂ L θ into intervalsof equal length and let χ θ,n be the expectation of χ θ with respect to P n . As ρ n → 0, thesequence of functions (χ θ,n ) n∈N converges pointwise to χ θ . By Fatou’s Lemma, we’re done if weprove that each χ θ,n is L 2 and its L 2 -norm ‖χ θ,n ‖ 2is bounded above by a constant independentof n.By definition,µ θ (J ρni) = m d((projθ ) −1 (J ρni) ) ≤ s ρn,i ·c 1 ·ρ n d , i = 1,2,...,⌊4/ρ n ⌋,and thenimplying thati)|J ρni|χ θ,n (x) = µ θ(J ρn‖χ θ,n ‖ 2 2==≤∫≤ c 1 ·s ρn,i ·ρ nd|J ρni|L θ|χ θ,n | 2 dm⌊4/ρ n⌋∑i=1⌊4/ρ n⌋∑i=1∫J ρni|J ρni|·|χ θ,n | 2 dm⌊4/ρ∑ n⌋2d−1 ≤ c 12 ·ρ n ·, ∀x ∈ J ρni,(c1 ·s ρn,i ·ρ ndi=1|J ρni|s ρn,i 2≤ c 12 ·ρ n2d−1 ·N (K1 ×K 2 ) ρn(θ).) 2In view of (2.4.2), this last expression is bounded above by( )·( )2d−1 1−2d c 12 ·ρ n c 3 ·ρ n ·m = c 12 ·c 3 ·m,which is a constant independent of n.

Chapter 2. Marstrand’s Theorem for products of Cantor sets 192.5 Concluding remarksThe proofs of Theorems 2.1.1 and 2.1.2 work not just for the case of products of regular Cantorsets, but in greater generality, whenever K ⊂ R 2 is a Borel set for which there is a constantc > 0 such that, for any x ∈ K and 0 ≤ r ≤ 1,c −1 ·r d ≤ m d (K ∩B r (x)) ≤ c·r d ,since this alone implies the existence of ρ-decompositions for K.

Chapter 2. Marstrand’s Theorem for products of Cantor sets 20

Chapter 3. Yet another proof of Marstrand’s Theorem 22sets.In 1954, J. M. Marstrand [27] proved the following result on the fractal dimension of planeTheorem 3.1.1. If K ⊂ R 2 is a Borel set such that HD(K) > 1, then m(proj θ (K)) > 0 form-almost every θ ∈ R.The proof is based on a qualitative characterization of the “bad” angles θ for which theresult is not true. Specifically, Marstrand exhibits a Borel measurable function f(x,θ), (x,θ) ∈R 2 ×[−π/2,π/2], such that f(x,θ) = ∞ for m s -almost every x ∈ K, for every “bad” angle. Inparticular,∫Kf(x,θ)dm s (x) = ∞. (3.1.1)On the other hand, using a version of Fubini’s Theorem, he proves that∫ π/2−π/2which, in view of (3.1.1), implies that∫dθ f(x,θ)dm s (x) = 0Km({θ ∈ [−π/2,π/2]; m(proj θ (K)) = 0}) = 0.These results are based on the analysis of rectangular densities of points.Many generalizations and simpler proofs have appeared since. One of them came in 1968 byR. Kaufman who gave a very short proof of Marstrand’s Theorem using methods of potentialtheory. See [21] for his original proof and [29], [37] for further discussion.In this chapter, we give a new proof of Theorem 3.1.1. Our proof makes a study on thefibers K ∩proj −1 θ (v), (θ,v) ∈ R×L θ , and relies on two facts:(I) Transversality condition: given two squares on the plane, the Lebesgue measure of the setof angles for which their projections have nonempty intersection has an upper bound. SeeSubsection 3.3.2.(II) After a regularization of K, (I) enables us to conclude that, except for a small set of anglesθ ∈ R, the fibers K ∩proj −1 θ (v) are not concentrated in a thin region. As a consequence, Kprojects into a set of positive Lebesgue measure.The idea of (II) is based on [31] used in [25] to develop a combinatorial proof of Theorem3.1.1 when K is the product of two regular Cantor sets. In the present chapter, we give acombinatorial proof of Theorem 3.1.1 without any restrictions on K. Compared to other proofsof Marstrand’s Theorem, the new ideas here are the discretization of the argument and the useof dyadic covers, which allow the simplification of the method employed. These covers may becomposed of sets with rather different scales and so a weighted sum is necessary to capture theHausdorff s-measure of K.

Chapter 3. Yet another proof of Marstrand’s Theorem 23The theory developed in [26] works whenever K is an Ahlfors-David regular set, namelywhen there are constants a,b > 0 such thata·r d ≤ m s (K ∩B r (x)) ≤ b·r d , for any x ∈ K and 0 < r ≤ 1.Unfortunately, the general situation can not be reduced to this one, as proved by P. Mattilaand P. Saaranen: in [30], they constructed a compact set of R with positive Lebesgue measuresuch that it contains no nonempty Ahlfors-David subset.We also give yet another proof that the push-forward measure of the restriction of m s to K,defined as µ θ = (proj θ ) ∗ (m s | K ), is absolutely continuous with respect to m, for m-almost everyθ ∈ R, and its Radon-Nykodim derivative is square-integrable.Theorem 3.1.2. The measure µ θ is absolutely continuous with respect to m and its Radon-Nykodim derivative is an L 2 function, for m-almost every θ ∈ R.Marstrand’s Theorem is a classical result in Geometric Measure Theory. In particular,if K = K 1 × K 2 is a cartesian product of two one-dimensional subsets of R, Marstrand’stheorem translates to “m(K 1 +λK 2 ) > 0 for m-almost every λ ∈ R”. The investigation of sucharithmetic sums K 1 + λK 2 has been an active area of Mathematics, in special when K 1 andK 2 are dynamically defined Cantor sets. Remarkably, M. Hall [18] proved, in 1947, that theLagrange spectrum 1 contains a whole half line, by showing that the arithmetic sum K(4)+K(4)of a certain Cantor set K(4) ⊂ R with itself contains [6,∞).In the connection of these two applications, we point out that a formula for the Hausdorffdimension of K 1 +K 2 , under mild assumptions of non-linear Cantor sets K 1 and K 2 , has beenobtainedbythesecondauthorin[31]andappliedin[32]toprovethattheHausdorffdimensionofthe Lagrange spectrum increases continuously. In parallel to this non-linear setup, Y. Peres andP. Shmerkin proved the same phenomena happen to self-similar Cantor sets without algebraicresonance [41]. Finally, M. Hochman and P. Shmerkin extended and unified many resultsconcerning projections of products of self-similar measures on regular Cantor sets [19].The content of this chapter is organized as follows. In Section 3.2 we introduce the basicnotations and definitions. Section 3.3 is devoted to the main calculations, including thetransversality condition in Subsection 3.3.2 and the proof of existence of good dyadic covers inSubsection 3.3.3. Finally, in Section 3.4 we prove Theorems 3.1.1 and 3.1.2. We also collectfinal remarks in Section 3.5.1 The Lagrange spectrum is the set of best constants of rational approximations of irrational numbers. See [9]for the specific description.

Chapter 3. Yet another proof of Marstrand’s Theorem 243.2 Preliminaries3.2.1 NotationThe distance in R 2 will be denoted by |·|. Let B r (x) denote the open ball of R 2 centered in xwith radius r. As in Section 3.1, the diameter of U ⊂ R 2 is |U| = sup{|x−y|;x,y ∈ U} and, ifU is a family of subsets of R 2 , the diameter of U is defined as‖U‖ = sup |U|.U∈UGiven s > 0, the Hausdorff s-measure of a set K ⊂ R 2 is m s (K) and its Hausdorff dimension isHD(K). In this work, we assume K is contained in [0,1) 2 .Definition 3.2.1. A Borel set K ⊂ R 2 is an s-set if HD(K) = s and 0 < m s (K) < ∞.Let m be the Lebesgue measure of Lebesgue measurable sets on R. For each θ ∈ R, letv θ = (cosθ,sinθ), L θ the line in R 2 through the origin containing v θ and proj θ : R 2 → L θ theorthogonal projection onto L θ .A square [a,a+l)×[b,b+l) ⊂ R 2 will be denoted by Q and its center, the point (a+l/2,b+l/2), by x.Let X be a set. We use the following notation to compare the asymptotic of functions.Definition 3.2.2. Let f,g : X → R be two real-valued functions. We say f g if there is aconstant C > 0 such thatIf f g and g f, we write f ∼ g.|f(x)| ≤ C ·|g(x)|, ∀x ∈ X.3.2.2 Dyadic squaresLet D 0 be the family of unity squares of R 2 congruent to [0,1) 2 and with vertices in the latticeZ 2 . Dilating this family by a factor of 2 −i , we obtain the family D i , i ∈ Z.Definition 3.2.3. Let D denote the union of D i , i ∈ Z. A dyadic square is any element Q ∈ D.The dyadic squares possess the following properties:(1) Every x ∈ R 2 belongs to exactly one element of each family D i .(2) Two dyadic squares are either disjoint or one is contained in the other.(3) A dyadic squareof D i is contained in exactly one dyadic squareof D i−1 and contains exactlyfour dyadic squares of D i+1 .

Chapter 3. Yet another proof of Marstrand’s Theorem 25(4) Given any subset U ⊂ R 2 , there are four dyadic squares, each with side length at most2·|U|, whose union contains U.(1) to (3) are direct. To prove (4), let R be smallest rectangle of R 2 with sides parallelto the axis that contains U. The sides of R have length at most |U|. Let i ∈ Z such that2 −i−1 < |U| ≤ 2 −i and choose a dyadic square Q ∈ D i that intersects R. If Q contains U, we’redone. If not, Q and three of its neighbors cover U.RQUDefinition 3.2.4. A dyadic cover of K is a finite subset C ⊂ D of disjoint dyadic squares suchthatK ⊂ ⋃ Q∈CQ.First used by A.S. Besicovitch in his demonstration that closed sets of infinite m s -measurecontainsubsetsofpositivebutfinitemeasure[5], dyadiccoverswerelateremployedbyMarstrandto investigate the Hausdorff measure of cartesian products of sets [28].Due to (4), for any family U of subsets of R 2 , there is a dyadic family C such that⋃Q ⊃ ⋃ ∑U and |Q| s < 64· ∑|U| sQ∈C U∈U Q∈C U∈Uand so, if K is an s-set, there exists a sequence (C i ) i≥1 of dyadic covers of K with diametersconverging to zero such that∑|Q| s ∼ 1. (3.2.1)Q∈C i3.3 CalculationsLet K ⊂ R 2 be a Borel set with Hausdorff dimension greater than one. From now on, weassume every cover of K is composed of dyadic squares of sides at most one. Before going intothe calculations, we make the following reduction.

Chapter 3. Yet another proof of Marstrand’s Theorem 26Lemma 3.3.1. Let K be a Borel subset of R 2 . Given s < HD(K), there exists a compact s-setK ′ ⊂ K such thatm s (K ′ ∩B r (x)) r d , x ∈ R 2 and 0 ≤ r ≤ 1.In other words, there exists a constant b > 0 such thatm s (K ′ ∩B r (x)) ≤ b·r d , for any x ∈ R 2 and 0 ≤ r ≤ 1. (3.3.1)See Theorem 5.4 of [11] for a proof of the above lemma when K is closed and [10] for thegeneral case. From now on, we assume K is a compact s-set, with s > 1, that satisfies (3.3.1).Given a dyadic cover C of K, let, for each θ ∈ [−π/2,π/2], fθ C : L θ → R be the functiondefined byfθ C (x) = Q∈Cχ ∑ projθ (Q)(x)·|Q| s−1 ,where χ projθ (Q) denotes the characteristic function of the set proj θ (Q). The reason we considerthis function is that it captures the Hausdorff s-measure of K in the sense that∫fθ C (x)dm(x) = ∑ ∫|Q| s−1 · χ projθ (Q)(x)dm(x)L θ Q∈CL θ= ∑ |Q| s−1 ·m(proj θ (Q))Q∈Cwhich, as |Q|/2 ≤ m(proj θ (Q)) ≤ |Q|, satisfies∫L θf C θ (x)dm(x) ∼ ∑ Q∈CIf in addition C satisfies (3.2.1), then∫(x)dm(x) ∼ 1 , ∀θ ∈ [−π/2,π/2]. (3.3.2)L θf C θDenoting the union ⋃ Q∈CQ by C as well, an application of the Cauchy-Schwarz inequality givesthatm(proj θ (C))·( ∫proj θ (C)(fCθ) 2dm)≥|Q| s .( ∫ ) 2fθ C dm ∼ 1.proj θ (C)The above inequality implies that if (C i ) i≥1 is a sequence of dyadic covers of K satisfying (3.2.1)with diameters converging to zero and the L 2 -norm of f C iθis uniformly bounded, that is∫ ( ) 2dm 1, (3.3.3)proj θ (C i )f C iθ

Chapter 3. Yet another proof of Marstrand’s Theorem 27thenm(proj θ (K)) = limi→∞m(proj θ (C i )) 1and so proj θ (K) has positive Lebesgue measure, as wished. This conclusion will be obtainedfor m-almost every θ ∈ [−π/2,π/2] by showing thatI i. =∫ π/2−π/2(dθ∫L θf C iθ) 2dm 1. (3.3.4)3.3.1 Rewriting the integral I iFor simplicity, let f denote f C iθ. Then the interior integral of (3.3.4) becomes∫L θf 2 dm =and, using the inequalitiesit follows that∫⎞ ⎛ ⎞⎝ ∑ χ projθ (Q) ·|Q| s−1 ⎠· ⎝ ∑˜Q∈Ci χ projθ (˜Q) ·|˜Q| s−1 ⎠dmQ∈C iL θ⎛= ∑ ∫|Q| s−1 ·|˜Q| s−1 · χ projθ (Q)∩proj θ (˜Q) dmLQ,˜Q∈C θi= ∑|Q| s−1 ·|˜Q| s−1 ·m(proj θ (Q)∩proj θ (˜Q))Q,˜Q∈C im(proj θ (Q)∩proj θ (˜Q)) ≤ min{m(proj θ (Q)),m(proj θ (˜Q)} ≤ min{|Q|,| ˜Q|},∫L θf 2 dm ∑Q,˜Q∈C i|Q| s−1 ·|˜Q| s−1 ·min{|Q|,| ˜Q|}. (3.3.5)We now proceed to prove (3.3.4) by a double-counting argument. To this matter, consider, foreach pair of squares (Q, ˜Q) ∈ C i ×C i , the setThenI i Θ Q,˜Q=∑{}θ ∈ [−π/2,π/2];proj θ (Q)∩proj θ (˜Q) ≠ ∅ .|Q| s−1 ·|˜Q| s−1 ·min{|Q|,|˜Q|}·Q,˜Q∈C i= ∑∫ π/2−π/2χ ΘQ, ˜Q(θ)dθQ,˜Q∈C i|Q| s−1 ·|˜Q| s−1 ·min{|Q|,|˜Q|}·m(Θ Q,˜Q). (3.3.6)

Chapter 3. Yet another proof of Marstrand’s Theorem 283.3.2 Transversality conditionThis subsection estimates the Lebesgue measure of Θ Q,˜Q.Lemma 3.3.2. If Q, ˜Q are squares of R 2 and x,˜x ∈ R 2 are its centers, respectively, thenProof. Let θ ∈ Θ Q,˜Qand consider the figure.( )m Θ Q,˜Q≤ 2π · max{|Q|,|˜Q|} ·|x− ˜x|xθ |θ −ϕ 0|˜xL θproj θ (˜x)θproj θ (x)Since proj θ (Q) has diameter at most |Q| (and the same happens to ˜Q), we have |proj θ (x) −proj θ (˜x)| ≤ 2·max{|Q|,|˜Q|} and then, by elementary geometry,sin(|θ −ϕ 0 |) = |proj θ(x)−proj θ (˜x)||x− ˜x|≤ 2· max{|Q|,|˜Q|}|x− ˜x|=⇒ |θ −ϕ 0 | ≤ π · max{|Q|,|˜Q|}|x− ˜x|because sin −1 y ≤ πy/2. As ϕ 0 is fixed, the lemma is proved.,We point out that, although ingenuous, Lemma 3.3.2 expresses the crucial property oftransversality that makes the proof work, and all results related to Marstrand’s Theorem use asimilar idea in one way or another. See [43] where this tranversality condition is also exploited.By Lemma 3.3.2 and (3.3.6), we obtainI i ∑|x− ˜x| −1 ·|Q| s ·|˜Q| s . (3.3.7)Q,˜Q∈C i

Chapter 3. Yet another proof of Marstrand’s Theorem 293.3.3 Good coversThe last summand will be estimated by choosing appropriate dyadic covers C i . Let C be anarbitrary dyadic cover of K. Remember K is an s-set satisfying (3.3.1).Definition 3.3.3. The dyadic cover C is good if∑|˜Q| s < max{128b,1}·|Q| s , ∀Q ∈ D. (3.3.8)˜Q∈C˜Q⊂QAny other constant dependingonly on K would work for the definition. Thereason we chosethis specific constant will become clear below, where we provide the existence of good dyadiccovers.Proposition 3.3.4. Let K ⊂ R 2 be a compact s-set satisfying (3.3.1). Then, for any δ > 0,there exists a good dyadic cover of K with diameter less than δ.Proof. Let i 0 ≥ 1 such that 2 −i 0−1 < δ ≤ 2 −i 0. Begin with a finite cover U of K with diameterless than δ/4 such that∑|U| s < 2·m s (K ∩Q) , ∀Q ∈ D i0 .U∈UU⊂QNow, change U by adyadic cover C according to property(4) of Subsection 3.2.1. C hasdiameterat most δ and satisfies∑|˜Q| s < 64· ∑|U| s < 128·m s (K ∩Q) , ∀Q ∈ D i0 .˜Q∈C˜Q⊂QU∈UU⊂QBy additivity, the same inequality happens for any Q ∈ ⋃ 0≤i≤i 0D i and so, as m s (K ∩ Q) ≤b·|Q| s , it follows that∑|˜Q| s < 128b·|Q| s , ∀Q ∈ ⋃˜Q∈C˜Q⊂Q0≤i≤i 0D i , (3.3.9)that is, (3.3.8) holds for large scales. To control the small ones, apply the following operation:whenever Q ∈ ⋃ i>i 0D i is such that∑|˜Q| s > |Q| s ,˜Q∈C˜Q⊂Qwechange C by C∪{Q}\{˜Q ∈ C; ˜Q ⊂ Q}. It is clear that suchoperation preserves theinequality(3.3.9) and so, after a finite number of steps, we end up with a good dyadic cover.

Chapter 3. Yet another proof of Marstrand’s Theorem 30As the constant in (3.3.8) does not depend on δ, there is a sequence (C i ) i≥1 of good dyadiccovers of K with diameters converging to zero such that∑|˜Q| s |Q| s , Q ∈ D and i ≥ 1. (3.3.10)˜Q∈C i˜Q⊂Q3.4 Proof of Theorems 3.1.1 and 3.1.2Proof of Theorem 3.1.1. Let (C i ) i≥1 beasequence of good dyadic covers satisfying (3.3.10) suchthat ‖C i ‖ → 0. By (3.3.7),I i ∑Q,˜Q∈C i|x− ˜x| −1 ·|Q| s ·|˜Q| s= ∑ Q∈C i ∞ ∑j=0≤ ∑ Q∈C i ∞ ∑j=0∑|x− ˜x| −1 ·|Q| s ·|˜Q| s˜Q∈C i2 −j−1 0, the setsG i ε = {θ ∈ [−π/2,π/2];(f∫L C iθθ) 2dm< ε−1}Then m ( [−π/2,π/2]\G i ε) ε, and the same holds for the setIf θ ∈ G ε , thenG ε = ⋂ i≥1∞⋃G j ε.j=im(proj θ (C i )) ε, for infinitely many n,, i ≥ 1.whichimpliesthatm(proj θ (K)) > 0. Finally, thesetG = ⋃ i≥1 G 1/i satisfiesm([−π/2,π/2]\G) =0 and m(proj θ (K)) > 0, for any θ ∈ G.

Chapter 3. Yet another proof of Marstrand’s Theorem 31A direct consequence is theCorollary 3.4.1. The measure µ θ = (proj θ ) ∗ (m s | K ) is absolutely continuous with respect tom, for m-almost every θ ∈ R.Proof. By Theorem 3.1.1, we have the implicationX ⊂ K, m s (X) > 0 =⇒ m(proj θ (X)) > 0, m-almost every θ ∈ R, (3.4.1)which is sufficient for the required absolute continuity. Indeed, if Y ⊂ L θ satisfies m(Y) = 0,thenµ θ (Y) = m s (X) = 0,where X = proj −1 θ (Y). Otherwise, by (3.4.1) we would have m(Y) = m(proj θ (X)) > 0,contradicting the assumption.Let f θ = dµ θ /dm. By the proof of Theorem 3.1.1, we have∥∥f C i∥θ∥L 2 1, m-a.e. θ ∈ R. (3.4.2)Proof of Theorem 3.1.2. Define, for each ε > 0, the function f θ,ε : L θ → R byf θ,ε (x) = 1 2ε∫ x+εx−εf θ (y)dm(y), x ∈ L θ .As f θ is an L 1 -function, the Lebesgue differentiation theorem gives that f θ (x) = lim ε→0 f θ,ε (x)for m-almost every x ∈ L θ . If we manage to show that 2‖f θ,ε ‖ L 2 1, m-a.e. θ ∈ R, (3.4.3)then Fatou’s lemma establishes the theorem. To this matter, first observe thatf θ,ε (x) = 1 2ε∫ x+εx−εf θ (y)dm(y)= 1 2ε ·µ θ([x−ε,x+ε])= 1 2ε ·m s((projθ ) −1 ([x−ε,x+ε])∩K ) .2 We consider ‖f θ,ε ‖ L 2 as a function of ε > 0.

Chapter 3. Yet another proof of Marstrand’s Theorem 32In order to estimate this last term, fix ε > 0 and let i > 0 such that C = C i has diameter lessthan ε. Thenm s((projθ ) −1 ([x−ε,x+ε])∩K ) ≤∼≤∑Q∈Cproj θ (Q)⊂[x−2ε,x+2ε]∑Q∈Cproj θ (Q)⊂[x−2ε,x+2ε]∑Q∈Cproj θ (Q)⊂[x−2ε,x+2ε]∫ x+2εx−2εm s (Q∩K)|Q| sf C θ (y)dm(y),|Q| s−1 ·m(proj θ (Q))where in the second inequality we used (3.3.1). By the Cauchy-Schwarz inequality,|f θ,ε (x)| 2 1 2ε∫ x+2εx−2ε∣ fCθ (y) ∣ 2 dm(y)and so‖f θ,ε ‖ 2 L 2∼∫∫2εL θ1L θ∣ ∣ f C θ= ∥ ∥ fCθ∥ ∥2L 2∫ x+2εx−2ε∣ 2 dm∣ fCθ (y) ∣ 2 dm(y)dm(x)which, by (3.4.2), establishes (3.4.3).3.5 Concluding remarksThe good feature of the proof is that the discretization idea may be applied to other contexts.For example, we prove in [24] a Marstrand type theorem in an arithmetical context.

Chapter 4A Marstrand theorem for subsets ofintegersWe prove a Marstrand type theorem for a class of subsets of the integers. More specifically,after defining the counting dimension D(E) of E ⊂ Z and the concepts of regularity andcompatibility, we show that if E,F ⊂ Z are two regular compatible sets, then D(E +⌊λF⌋) ≥min{1,D(E)+D(F)} for Lebesgue almost every λ ∈ R. If in addition D(E)+D(F) > 1, it isalso shown that E+⌊λF⌋ has positive upper Banach density for Lebesgue almost every λ ∈ R.The result has direct consequences when applied to arithmetic sets, such as the integer valuesof a polynomial with integer coefficients.4.1 IntroductionThe purpose of this chapter is to prove a Marstrand type theorem for a class of subsets of theintegers.Thewell-known theorem of Marstrand [27] states the following: if K ⊂ R 2 is a Borel set suchthat its Hausdorff dimension is greater than one then, for almost every direction, its projectionto R in the respective direction has positive Lebesgue measure. In other words, this means K is“fat” in almost every direction. When K is the product of two real subsets K 1 ,K 2 , Marstrand’stheorem can be stated in a more analytical form as: for Lebesgue almost every λ ∈ R, thearithmetic sum K 1 +λK 2 has positive Lebesgue measure. Much research has been done aroundthis topic, specially because the analysis of such arithmetic sums has applications in the theoryof homoclinic birfurcations and also in diophantine approximations.Given a subset E ⊂ Z, let d ∗ (E) denote its upper Banach density 1 . A remarkable result in1 See Subsection 4.2.2 for the proper definitions.33

Chapter 4. Marstrand theorem for subsets of integers 34additive combinatorics is Szemerédi’s theorem [45]. It asserts that if d ∗ (E) > 0, then E containsarbitrarily long arithmetic progressions. One can interpret this result by saying that densityrepresents the correct notion of largeness needed to preserve finite configurations of Z. On theother hand, Szemerédi’s theorem cannot infer any property about subsets of zero upper Banachdensity, and many of these sets are of interest. A class of examples are the integer values of apolynomial with integer coefficients and the prime numbers. These sets may, as well, containcombinatorially rich patterns. See, for example, [4] and [16].A set E ⊂ Z of zero upper Banach density is characterized as occupying portions in intervalsof Z that grow in a sublinear fashion as the length of the intervals grow. On the other hand,there still may exist some kind of growth speed. For example, the number of perfect squares onan initial interval of length n is about n 0.5 . This exponent means, in some sense, a dimensionof {n 2 ; n ∈ Z} inside Z. In the present chapter, we suggest a counting dimension D(E) for Eand establish the following Marstrand type result on the counting dimension of most arithmeticsums E +⌊λF⌋ for a class of subsets E,F ⊂ Z.Theorem 4.1.1. Let E,F ⊂ Z be two regular compatible sets. ThenD(E +⌊λF⌋) ≥ min{1,D(E) +D(F)}for Lebesgue almost every λ ∈ R. If in addition D(E)+D(F) > 1, then E +⌊λF⌋ has positiveupper Banach density for Lebesgue almost every λ ∈ R.An immediate consequence of Theorem 4.1.1 is that the same result holds for subsets of thenatural numbers.Corollary 4.1.2. Let E,F ⊂ N be two regular compatible sets. ThenD(E +⌊λF⌋) ≥ min{1,D(E) +D(F)}for Lebesgue almost every λ > 0. If in addition D(E)+D(F) > 1, then E +⌊λF⌋ has positiveupper Banach density for Lebesgue almost every λ > 0.The reader should make a parallel between the quantities d ∗ (E) and D(E) of subsets E ⊂ Zand Lebesgue measure and Hausdorff dimension of subsets of R. It is exactly this associationthat allows us to call Theorem 4.1.1 a Marstrand theorem for subsets of integers.The notions of regularity and compatibility are defined in Subsections 4.4.1 and 4.4.2, respectively.Both are fulfilled for many arithmetic subsets of Z, such as the integer values of apolynomial with integer coefficients and, more generally, for the universal sets (see Definition4.4.5). These are the sets that exhibit the expected growth rate along intervals of arbitrarylength. The second result of this work is

Chapter 4. Marstrand theorem for subsets of integers 35Theorem 4.1.3. Let E 0 ,...,E k be universal subsets of Z. ThenD(E 0 +⌊λ 1 E 1 ⌋+···+⌊λ k E k ⌋) ≥ min{1,D(E 0 )+D(E 1 )+···+D(E k )}for Lebesgue almost every λ = (λ 1 ,...,λ k ) ∈ R k . If in addition ∑ ki=0 D(E i) > 1, then thearithmetic sum E 0 + ⌊λ 1 E 1 ⌋ + ··· + ⌊λ k E k ⌋ has positive upper Banach density for Lebesguealmost every λ = (λ 1 ,...,λ k ) ∈ R k .The integer values of a polynomial with integer coefficients have special interest in ergodictheory and its connections with combinatorics, due to ergodic theorems along these subsets [6],as well as its combinatorial implications. See [4] for the remarkable work of V. Bergelson andA. Leibman on the polynomial extension of Szemerédi’s theorem. Theorem 4.1.3 has a directconsequence when applied to this setting.Corollary 4.1.4. Let p i ∈ Z[x] with degree d i > 0 and let E i = (p i (n)) n∈Z , i = 0,...,k. ThenD(E 0 +⌊λ 1 E 1 ⌋+···+⌊λ k E k ⌋) ≥ min{1, 1 + 1 +···+ 1 }d 0 d 1 d kfor Lebesgue almost every λ = (λ 1 ,...,λ k ) ∈ R k . If in addition ∑ ki=0 d i −1 > 1, then thearithmetic sum E 0 + ⌊λ 1 E 1 ⌋ + ··· + ⌊λ k E k ⌋ has positive upper Banach density for Lebesguealmost every λ = (λ 1 ,...,λ k ) ∈ R k .The proof of Theorem 4.1.1 is based on the ideas developed in [24] and [26]. It relies onthe fact that the cardinality of a regular subset of Z along an increasing sequence of intervalsexhibits an exponential behavior ruled out by its counting dimension. As this holds for tworegular subsets E,F ⊂ Z, the compatibility assumption allows to estimate the cardinality ofthe arithmetic sum E + ⌊λF⌋ along the respective arithmetic sums of intervals and, finally, adouble-counting argument estimates the size of the “bad” parameters for which such cardinalityis small. Theorem 4.1.3 follows from Theorem 4.1.1 by a fairly simple induction.This chapter is organized as follows. In Section 4.2 we introduce the basic notations anddefinitions. Section 4.3 is devoted to the discussion of examples. In particular, the sets given byinteger values of a polynomial with integer coefficients are investigated in Subsection 4.3.1. InSection 4.4 we introduce the notions of regularity and compatibility. Subsection 4.4.3 provides acounterexample to Theorem 4.1.1 when the sets are no longer compatible and Subsection 4.4.4a counterexample to the same theorem when the space of parameters is Z. Finally, in Section4.5 we prove Theorems 4.1.1 and 4.1.3. We also collect final remarks and further questions inSection 4.6.

Chapter 4. Marstrand theorem for subsets of integers 364.2 Preliminaries4.2.1 General notationGiven a set X, |X| denotes the cardinality of X. Z denotes the set of integers and N the set ofpositive integers. We use the following notation to compare the asymptotic of functions.Definition 4.2.1. Let f,g : Z or N → R be two real-valued functions. We say f g if there isa constant C > 0 such that|f(n)| ≤ C ·|g(n)|, ∀n ∈ Z or N.If f g and g f, we write f ∼ g. We say f ≈ g iff(n)lim|n|→∞ g(n) = 1.For each x ∈ R, ⌊x⌋ denotes the integer part of x. For each k ≥ 1, m k denotes the Lebesguemeasure of R k . For k = 1, let m = m 1 . The letter I will always denote an interval of Z:I = (M,N] = {M +1,...,N}.The length of I is equal to its cardinality, |I| = N −M.For E ⊂ Z and λ ∈ R, λE denotes the set {λn; n ∈ E} ⊂ R and ⌊λE⌋ the set {⌊λn⌋; n ∈E} ⊂ Z.4.2.2 Counting dimensionDefinition 4.2.2. The upper Banach density of E ⊂ Z is equal towhere I runs over all intervals of Z.d ∗ (E) = limsup|I|→∞|E ∩I||I|Definition 4.2.3. The counting dimension or simply dimension of a set E ⊂ Z is equal towhere I runs over all intervals of Z.D(E) = limsup|I|→∞,log|E ∩I|log|I|Obviously, D(E) ∈ [0,1] and D(E) = 1 whenever d ∗ (E) > 0. The counting dimension allowsus to distinguish largeness between two sets of zero upper Banach density. Similar definitions toD(E) have appeared in [2] and [13]. We now give another characterization to it that is similarin spirit to the Hausdorff dimension of subsets of R. Let α be a nonnegative real number.,

Chapter 4. Marstrand theorem for subsets of integers 37Definition 4.2.4. The counting α-measure of E ⊂ Z is equal towhere I runs over all intervals of Z.H α (E) = limsup|I|→∞|E ∩I||I| α ,Clearly, H α (E) ∈ [0,∞]. For a fixed E ⊂ Z, the numbers H α (E) are decreasing in α andone easily checks thatα < D(E) =⇒ H α (E) = ∞ and α > D(E) =⇒ H α (E) = 0,which in turn implies the existence and uniqueness of α ≥ 0 such thatH β (E) = ∞ , if 0 ≤ β < α,= 0 , if β > α.The above equalities imply that D(E) = α, that is, the counting dimension is exactly theparameter α where H α (E) decreases from infinity to zero. Also, if β > D(E), then|E ∩I| |I| β , (4.2.1)where I runs over all intervals of Z and, conversely, if (4.2.1) holds, then D(E) ≤ β.Below, we collect basic properties of the counting dimension and α-measure.(i) E ⊂ F =⇒ D(E) ≤ D(F).(ii) D(E ∪F) = max{D(E),D(F)}.(iii) For any λ > 0,H α (⌊λE⌋) = λ −α ·H α (E). (4.2.2)The first two are direct. Let’s prove (iii). For any interval I ⊂ Z, we have |E ∩ I| ≈|⌊λE⌋∩⌊λI⌋| and so|E ∩I||I| α≈ λ α · |⌊λE⌋∩⌊λI⌋||⌊λI⌋| α=⇒ H α (E) = λ α ·H α (⌊λE⌋).Remark 4.2.5. As ⌊−x⌋ = −⌊x⌋ or ⌊−x⌋ = −⌊x⌋ − 1, the sets ⌊−λE⌋ and ⌊λE⌋ have thesame counting dimension. Also, 0 < H α (⌊−λE⌋) < ∞ if and only if 0 < H α (⌊λE⌋) < ∞. Forthese reasons, we assume from now on that λ > 0.

Chapter 4. Marstrand theorem for subsets of integers 384.3 ExamplesExample 1. Let α ∈ (0,1] andE α ={⌊n 1/α⌋ }; n ∈ N . (4.3.1)We infer that H α (E α ) = 1. To prove this, we make use of the inequality 2(x+y) α ≤ x α +y α , x,y ≥ 0,to conclude that|E α ∩(M,N]|(N −M) α ≤ (N +1)α −(M +1) α(N −M) α ≤ 1.This proves that H α (E α ) ≤ 1. On the other hand,|E α ∩(0,N]|N α≥ Nα −1N αand so H α (E α ) ≥ 1. Then H α (E α ) = 1 and, in particular, D(E α ) = α.Example 2. The setE = ⋃ n∈N(n n ,(n+1) n ]∩E 1−1/nhas zero upper Banach density and D(E) = 1.Example 3. The prime numbers have dimension one. This follows from the prime numbertheorem:log|{1 ≤ p ≤ n; p is prime}| logn−loglognlim= lim = 1.n→∞ logn n→∞ lognExample 4. Sets of zero upper Banach density appear naturally in infinite ergodic theory. Let(X,A,µ,T) be a sigma-finite measure-preserving system, with µ(X) = ∞, and let A ∈ A havefinite measure. Fixed x ∈ A, let E = {n ≥ 1; T n x ∈ A}. By Hopf’s Ratio Ergodic Theorem,E has zero upper Banach density almost surely. In many specific cases, its dimension can becalculated or at least estimated. See [1] for further details.4.3.1 Polynomial subsets of ZDefinition 4.3.1. A polynomial subset of Z is a set E = {p(n); n ∈ Z}, where p(x) is anon-constant polynomial with integer coefficients.2 For each t ≥ 0, the function x ↦→ x α +(t−x) α , 0 ≤ x ≤ t, is concave and so attains its minimum in x = 0and x = t. Then x α +(t−x) α ≥ t α for any x ∈ [0,t].

Chapter 4. Marstrand theorem for subsets of integers 39These are the sets we consider in Theorem 4.1.3. Their counting dimension is easily calculatedasfollows.GivenE,F ⊂ Z,letE andF beasymptoticifE = {··· < a −1 < a 0 < a 1 < ···},F = {··· < b −1 < b 0 < b 1 < ···} and there is i ≥ 0 such thata n−i ≤ b n ≤ a n+i , for every n ∈ Z. (4.3.2)Denote this by E ≈ F.Lemma 4.3.2. If E,F ⊂ Z are asymptotic then H α (E) = H α (F), for any α ≥ 0. In particular,D(E) = D(F).Proof. Let I = (M,N] be an interval and assume E ∩I = {a m+1 ,a m+2 ,...,a n }. By relation(4.3.2),b m−i ≤ a m ≤ M < a m+1 ≤ b m+i+1 and b n−i ≤ a n ≤ N < a n+1 ≤ b n+i+1 ,which imply the inclusions{b m+i+1 ,...,b n−i } ⊂ F ∩I ⊂ {b m−i ,...,b n+i+1 }.Then |E ∩I| ≈ |F ∩I| and soH α (E) = limsup|I|→∞|E ∩I||I| α= limsup|I|→∞|F ∩I||I| α= H α (F).Let E = {p(n); n ∈ Z}, where p(x) ∈ Z[x] has degree d. Assuming p has leading coefficienta > 0, there is i ≥ 0 such that a · (n − i) d < p(n) < a · (n + i) d for every n ∈ Z and thenE ≈ aE 1/d , where E 1/d is defined as in (4.3.1). By Lemma 4.3.2, we getD(E) = 1 d and H 1/d(E) = 1.4.3.2 Cantor sets in ZThe famous ternary Cantor set of R is formed by the real numbers of [0,1] with only 0’s and2’s on their base 3 expansion. In analogy to this, let E ⊂ Z be defined as{ n∑}E = a i ·3 i ; n ∈ N and a i = 0 or 2 . (4.3.3)i=0The set E has been slightly investigated in [11]. There, A. Fisher proved thatH log2/log3 (E) > 0.

Chapter 4. Marstrand theorem for subsets of integers 40We prove below that H log2/log3 (E) ≤ 2, which in particular gives that D(E) = log2/log3,as expected. Let I = (M,N] be an interval of Z. We can assume M + 1,N ∈ E. Indeed, ifĨ = ( ˜M,Ñ], where ˜M +1 and Ñ are the smallest and largest elements of E ∩I, respectively,thenLet M +1,N ∈ E, say|E ∩I||I| α≤|E ∩Ĩ||Ĩ|α ·M +1 = a 0 ·3 0 +a 1 ·3 1 +···+a m−1 ·3 m +2·3 mN = b 0 ·3 0 +b 1 ·3 1 +···+b n−1 ·3 n +2·3 n .We can also assume that m < n. If this is not the case, the quotient |E ∩ I|/|I| α does notdecrease if we change I by (M −2·3 n ,N −2·3 n ]. With these assumptions,and thenN −M ≥ 2·3 n −(3 m+1 −2) ≥ 3 n|E ∩I| |E ∩(0,N]|≤|I| log2/log3 |I| log2/log3 ≤ 2 n+1(3 n = 2.) log2/log3Because I is arbitrary, this gives that H log2/log3 (E) ≤ 2.Observe that the renormalization of E ∩ (0,3 n ) via the linear map x ↦→ x/3 n generatesa subset of the unit interval (0,1) that is equal to the set of left endpoints of the remainingintervals of the n-th step of the construction of the ternary Cantor set of R. In other words,if K = ⋃ n∈E [n,n + 1], then K/3n is exactly the n-th step of the construction of the ternaryCantor set of R.More generally, let us define a class of Cantor sets in Z. Fix a basis a ∈ N and a binarymatrix A = (a ij ) 0≤i,j≤a−1 . LetΣ n (A) = { (d 0 d 1···d n−1 d n ); a di−1 d i= 1, 1 ≤ i ≤ n } , n ≥ 0,denote the set of admissible words of length n+1 and Σ ∗ (A) = ⋃ n≥0 Σ n(A) the set of all finiteadmissible words.Definition 4.3.3. The integer Cantor set E A ⊂ Z associated to the matrix A is the setE A = {d 0 ·a 0 +···+d n ·a n ; (d 0 d 1···d n ) ∈ Σ ∗ (A)}.Ourdefinition was inspiredon thefact that dynamically defined topologically mixingCantorsets of the real line are homeomorphic to subshifts of finite type, which is exactly what we didabove, after truncating the numbers. See [24] for more details. The dimension of E A , as in the

Chapter 4. Marstrand theorem for subsets of integers 41inspiring case, depends on the Perron-Frobenius eigenvalue of A. Remember that the Perron-Frobenius eigenvalue is the largest eigenvalue λ + (A) of A. It has multiplicity one and maximizesthe absolute value of the eigenvalues of A. Also, there is a constant c = c(A) > 0 such thatwhose proof may be found in [22].c −1 ·λ + (A) n ≤ |Σ n (A)| ≤ c·λ + (A) n , for every n ≥ 0, (4.3.4)Lemma 4.3.4. If A is a binary a×a matrix, thenD(E A ) = logλ +(A)logaand 0 < Hlogλ + (A)(E A ) < ∞.logaProof. Let I = (M,N]. Again, we may assume M +1,N ∈ E A , sayM +1 = x 0 ·a 0 +···+x n ·a nN = y 0 ·a 0 +···+y n ·a n ,with y n > x n . If y n ≥ x n +2, then{M +1 ≤ (x n +1)·a nand, as I ⊂ (0,a n+1 ), we have=⇒ |I| ≥ anN ≥ (x n +2)·an |E A ∩I||I| logλ + (A)loga≤ |Σ n(A)|a nlogλ + (A)loga≤ c·λ +(A) nλ + (A) n = c. (4.3.5)If y n = x n +1, let i,j ∈ {0,1,...,n−1} be the indices for which(i) x i < a−1 and x i+1 = ··· = x n−1 = a−1,(ii) y j > 0 and y j+1 = ··· = y n−1 = 0.Then {M +1 ≤ (x n +1)·a n −a iN ≥ (x n +1)·a n +a j =⇒ |I| ≥ a i +a j ≥ a max{i,j} .In order to ∑ nk=0 z k ·a k belong to I, one must have z n = x n or z n = x n +1. In the first casez i+1 = ··· = z n−1 = a−1 and in the second z j+1 = ··· = z n−1 = 0. Then|E A ∩I| ≤ |Σ i (A)|+|Σ j (A)| ≤ 2c·λ + (A) max{i,j}and so|E A ∩I||I| logλ + (A)loga≤ 2c·λ +(A) max{i,j}= 2c. (4.3.6)a max{i,j}logλ + (A)loga

Chapter 4. Marstrand theorem for subsets of integers 42Estimates (4.3.5) and (4.3.6) give H logλ+ (A)/loga < ∞. On the other hand,|E A ∩(0,a n ]|a nlogλ + (A)logawhich concludes the proof.≥ c−1 ·λ + (A) n−1λ + (A) n = c −1 ·λ + (A) −1 =⇒ H logλ+ (A)/loga > 0,By the above lemma, if X ⊂ {0,...,a−1} and A = (a ij ) is defined by a ij = 1 iff i,j ∈ X,then D(E A ) = log|X|/loga, which extends the results about the ternary Cantor set defined in(4.3.3).Ingeneral, ifE,F aresubsetsofZsuchthatD(E)+D(F) > 1, itisnottruethatd ∗ (E+F) >0, because the elements of E +F may have many representations as the sum of one element ofE and other of F. This resonance phenomenon decreases the dimension of E+F. Lemma 4.3.4provides a simple example to this situation: if E = E A and F = E B , where A = (a ij ) 0≤i,j≤11 ,B = (b ij ) 0≤i,j≤11 are defined bya ij = 1 ⇐⇒ 0 ≤ i,j ≤ 3 and b ij = 1 ⇐⇒ 4 ≤ i,j ≤ 7,then D(E)+D(F) = 2log4/log12, while E +F = E C for C = (c ij ) 0≤i,j≤11 given byc ij = 1 ⇐⇒ 4 ≤ i,j ≤ 10.E+F has counting dimension equal to log7/log12 and so d ∗ (E+F) = 0. What Theorem 4.1.1proves is that resonance is avoided if one is allowed to change the scales of the sets, multiplyingone of them by a factor λ ∈ R.4.3.3 Generalized IP-setsThis class of sets was suggested to us by Simon Griffiths and Rob Morris.Definition 4.3.5. Thegeneralized IP-setassociatedtothesequencesofpositiveintegers(k n ) n≥1 ,(d n ) n≥1 is the set { n∑}x i ·d i ; n ∈ N and 0 ≤ x i < k ii=1E =.We assume d n > ∑ n−1i=1 k i·d i , in which case the map ∑ ni=1 x i·d i ↦→ (x 1 ,...,x n ) is a bijectionbetween E and the set of finite sequencesΣ ∗ = {(x 1 ,...,x n ); n ∈ N,x n > 0 and 0 ≤ x i < k i }.Also, if Σ ∗ is lexicographically ordered 3 , then the map is order-preserving.3 The sequence (x 1,...,x n) is smaller than (y 1,...,y m) if n < m or if there is i ∈ {1,...,n} such that x i < y iand x j = y j for j = i+1,...,n.

Chapter 4. Marstrand theorem for subsets of integers 43Lemma 4.3.6. Let E be the generalized IP-set associated to (k n ) n≥1 , (d n ) n≥1 and let p n =k 1···k n . Thenlimsupn→∞logp n logp n≤ D(E) ≤ limsup · (4.3.7)logk n d n n→∞ logd nIn particular, if logk n /logp n → 0, then D(E) = limsup n→∞ logp n /logd n .Proof. The calculations are similar to those of Lemma 4.3.4. Let I = (M,N], sayM +1 = x 1 ·d 1 +···+x n ·d nN = y 1 ·d 1 +···+y n ·d n ,with y n > x n . If y n ≥ x n +2, then{M +1 ≤ (x n +1)·d nN ≥ (x n +2)·d n=⇒ |I| ≥ d nand |E ∩I| ≤ p n , thus givinglog|E ∩I|log|I|If y n = x n +1, let i,j ∈ {1,...,n−1} be the indices for which(i) x i < k i −1 and x j = k j −1, j = i+1,...,n−1,(ii) y j > 0 and y j+1 = ··· = y n−1 = 0.Then≤ logp nlogd n· (4.3.8)n−1∑n−1∑M +1 ≤ x n ·d n +x i ·d i + k l ·d l < x n ·d n −d i + k l ·d l < (x n +1)·d n −d il=1l≠il=1andN ≥ y j ·d j +y n ·d n ≥ (x n +1)·d n +d j ,implying that |I| ≥ d i +d j ≥ d max{i,j} . Now, in order to ∑ nl=1 z l·d l belong to I, one must havez n = x n or z n = x n +1. In the first case z i+1 = k i+1 −1,...,z n−1 = k n−1 −1 and in the secondz j+1 = ··· = z n−1 = 0. Then |E ∩I| ≤ p i +p j ≤ 2p max{i,j} and solog|E ∩I|log|I|≤ log2p max{i,j}logd max{i,j}· (4.3.9)Relations (4.3.8) and (4.3.9) prove the right hand inequality of (4.3.7). The other follows byconsidering the intervals (0, ∑ ni=1 k i ·d i ].

Chapter 4. Marstrand theorem for subsets of integers 444.4 Regularity and compatibility4.4.1 Regular setsDefinition 4.4.1. A subset E ⊂ Z is called regular or D(E)-set if 0 < H D(E) (E) < ∞.By Lemmas 4.3.2 and 4.3.4, polynomial sets and Cantor sets are regular.Definition 4.4.2. Given two subsets E = {··· < x −1 < x 0 < x 1 < ···} and F of Z, let E ∗Fdenote the setE ∗F = {x n ; n ∈ F}.This is a subset of E whose counting dimension is at most D(E)D(F). To see this, consideran arbitrary interval I ⊂ Z. If E ∩I = {x i+1 ,...,x j }, then(E ∗F)∩I = {x n ;n ∈ F ∩(i,j]}.Given α > D(E) and β > D(F), relation (4.2.1) guarantees that|(E ∗F)∩I| = |F ∩(i,j]| (j −i) β = |E ∩I| β |I| αβand so D(E ∗F) ≤ αβ. Choosing α, β arbitrarily close to D(E), D(F), respectively, it followsthat D(E ∗F) ≤ D(E)D(F).If E is regular, it is possible to choose F with arbitrary dimension in such a way that E∗Fis also regular and has dimension equal to D(E)D(F). To this matter, choose disjoint intervalsI n = (a n ,b n ], n ≥ 1, with lengths going to infinity such that|E ∩I n ||I n | D(E) 1and let E ∩I n = {x in+1 < x in+2 < ··· < x jn }, where i n < j n . Given α ∈ [0,1], letF = ⋃ n≥1(E α +i n )∩(i n ,j n ],where E α is defined as in (4.3.1). Then D(F) = D(E α ) = α and|(E ∗F)∩I n | ≥ |F ∩(i n ,j n ]|= |E α ∩(0,j n −i n ]| (j n −i n ) D(F)= |E ∩I n | D(F) |I n | D(E)D(F) ,

Chapter 4. Marstrand theorem for subsets of integers 45implying that|(E ∗F)∩I n | |I n | D(E)D(F) . (4.4.1)This proves the reverse inequality D(E∗F) ≥ D(E)D(F). We thus obtain that, given a regularsubset E ⊂ Z and 0 ≤ α ≤ D(E), there exists a regular subset E ′ ⊂ E such that D(E ′ ) = α. Itis a harder task to prove that this holds even when E is not regular.Proposition 4.4.3. Let E ⊂ Z and 0 ≤ α ≤ 1. If H α (E) > 0, then there exists a regular subsetE ′ ⊂ E such that D(E ′ ) = α. In particular, for any 0 ≤ α < D(E), there is E ′ ⊂ E regularsuch that D(E ′ ) = α.Proof. The idea is to apply a dyadic argument in E to decrease H α (E) in a controlled way.Given an interval I ⊂ Z and a subset F ⊂ Z, defines F (I) = . |F ∩J|supJ⊂I |J| α ·J intervalIf F = {a 1 ,a 2 ,...,a k } ⊂ I, the dyadic operation of alternately discard the elements a 2 ,a 4 ,...,a 2⌊(k−1)/2⌋ of the set {a 2 ,a 3 ,...,a k−1 },F = {a 1 ,a 2 ,...,a k } F ′ = {a 1 ,a 3 ,a 5 ,...,a 2⌈(k−1)/2⌉−1 ,a k },decreases s F (I) to approximately s F (I)/2. More specifically, if s F (I) > 2, then s F ′(I) > 1/2.Indeed, for every interval J ⊂ I|F ′ ∩J||J| αand, for J maximizing s F (I),≤ 1 2· |F ∩J|+1|J| α≤ s F(I)2+ 1 2 < s F(I)− 1 2|F ′ ∩J||J| α≥ 1 2· |F ∩J|−1|J| α> 1−12·|J| α ≥ 1 2 ·After a finite number of these dyadic operations, one obtains a subset F ′ ⊂ F such that12 < s F ′(I) ≤ 2.If H α (E) < ∞, there is nothing to do. Assume that H α (E) = ∞. We proceed inductivelyby constructing a sequence F 1 ⊂ F 2 ⊂ ··· of finite subsets of E contained in a sequence ofintervals I n = (a n ,b n ] with increasing lengths such that(i) 1/2 < s Fn (I n ) ≤ 3;(ii) there is an interval J n ⊂ I n such that |J n | ≥ n and|F n ∩J n ||J n | α > 1 2 ·

Chapter 4. Marstrand theorem for subsets of integers 46Once these properties are fulfilled, the set E ′ = ⋃ n≥1 F n will satisfy the required conditions.Take any a ∈ E and I 1 = {a}. Assume I n , F n and J n have been defined satisfying (i) and(ii). As H α (E) = ∞, there exists an interval J n+1 disjoint from (a n −|I n | 1/α ,b n +|I n | 1/α ] forwhich|E ∩J n+1 ||J n+1 | α ≥ (n+1) 1−α .This inequality allows to restrict J n+1 to a smaller interval of size at least n+1, also denotedJ n+1 , such thats E (J n+1 ) = |E ∩J n+1||J n+1 | α · (4.4.2)Consider F ′ n+1 = E ∩J n+1 and apply the dyadic operation to F ′ n+1 until12 < s F n+1 ′ (J n+1) = |F′ n+1 ∩J n+1||J n+1 | α ≤ 2. (4.4.3)Let I n+1 = I n ∪K n ∪J n+1 be the convex hull 4 of I n and J n+1 and F n+1 = F n ∪F ′ n+1 . Condition(ii) is satisfied because of (4.4.3). To prove (i), let I be a subinterval of I n+1 . We have threecases to consider.• I ⊂ I n ∪K n : by condition (i) of the inductive hypothesis,• I ⊂ K n ∪J n+1 : by (4.4.3),|F n+1 ∩I||I| α ≤ |F n ∩(I ∩I n )||I ∩I n | α ≤ 3.|F n+1 ∩I||I| α≤ |F′ n+1 ∩(I ∩J n+1)||I ∩J n+1 | α ≤ 2.• I ⊃ K n : as |K n | ≥ |I n | 1/α ,|F n+1 ∩I||I| α= |F n ∩(I ∩I n )|+|F ′ n+1 ∩(I ∩J n+1)|(|I ∩I n |+|K n |+|I ∩J n+1 |) α≤|F n ∩(I ∩I n )|(|I ∩I n |+|K n |) α + |F′ n+1 ∩(I ∩J n+1)|(|I ∩J n+1 |) α≤ |I n||K n | α +s F ′ n+1 (J n+1)≤ 3.This proves condition (i) and completes the inductive step.By the above proposition and equation (4.2.2), for any α ∈ [0,1] and h ∈ (0,∞), there existsan α-set E ⊂ Z such that H α (E) = h. We’ll use this fact in Subsection 4.4.3.4 Observe that |K n| ≥ |I n| 1/α .

Chapter 4. Marstrand theorem for subsets of integers 474.4.2 Compatible setsDefinition 4.4.4. Two regular subsets E,F ⊂ Z are compatible if there exist two sequences(I n ) n≥1 , (J n ) n≥1 of intervals with increasing lengths such that1. |I n | ∼ |J n |,2. |E ∩I n | |I n | D(E) and |F ∩J n | |J n | D(F) .The notion of compatibility means that E and F have comparable intervals on which therespective intersections obey the correct growth speed of cardinality. Some sets have theseintervals in all scales.Definition 4.4.5. A regular subset E ⊂ Z is universal if there exists a sequence (I n ) n≥1 ofintervals such that |I n | ∼ n and |E ∩I n | |I n | D(E) .It is clear that E and F are compatible whenever one of them is universal and the otheris regular. Every E α is universal and the same happens to polynomial subsets E, due to theasymptotic relation E ≈ aE 1/d , where d is the degree of the polynomial that defines E (seeSubsection 4.3.1). In particular, any two polynomial subsets are compatible.4.4.3 A counterexample of Theorem 4.1.1 for regular non-compatible setsIn this subsection, we construct regular sets E,F ⊂ Z such that D(E)+D(F) > 1 and E+⌊λF⌋has zero upper Banach density for every λ ∈ R. The idea is, in the contrast to compatibility,|E ∩I| |F ∩J|construct E and F such that the intervals I,J ⊂ Z for which and are bounded|I| D(E) |J| D(F)away from zero have totally different sizes.Let α ∈ (1/2,1). We will defineE = ⋃i=1,3,...such that the following conditions hold:(E i ∩I i ) and F = ⋃i=2,4,...(E i ∩I i )(i) E i = ⌊µ i ⌊µ i −1 E 0 ⌋⌋, i ≥ 1, where E 0 ⊂ N is an α-set with H α (E 0 ) = 1/2.(ii) I i = (a i ,b i ], i ≥ 1, is a disjoint sequence of intervals of increasing length such that|E i ∩I i |limi→∞ |I i | α = 1 2 ·(iii) µ i > b i−121−α for every i ≥ 1.

Chapter 4. Marstrand theorem for subsets of integers 48It is clear that we can inductively construct the sequences (µ i ) i≥1 ,(E i ) i≥1 and (I i ) i≥1 in sucha way that 0 < b i < a i+1 for every i ≥ 1. Before going into the calculations, let us explain theabove conditions. Thedilations in(i) guarantee that consecutive elements of E i differ at least by(the order of) µ i ; (ii) ensures that E,F are α-sets; (iii) ensures that the referred incompatibilitybetween E and F holds and, also, that the left endpoint of I i is much bigger than the rightendpoint of I i−1 .Lemma 4.4.6. Let E,F ⊂ Z with D(E),D(F) < 1, A,B ⊂ Z finite and E ′ = E ∪ A,F ′ = F ∪B. ThenProof. We haved ∗ (E ′ +⌊λF ′ ⌋) = d ∗ (E +⌊λF⌋), ∀λ ∈ R.E ′ +⌊λF ′ ⌋ = (E +⌊λF⌋)∪(A+⌊λF⌋)∪(E +⌊λB⌋)∪(A+⌊λB⌋)and each of the sets A+⌊λF⌋, E +⌊λB⌋, A+⌊λB⌋ has dimension smaller than one.Fix λ > 0. By removing, if necessary, finitely many intervals I i , we may also assume that(iv) ⌊λI i ⌋∩⌊λI j ⌋ = ∅ for all i ≠ j.(v) b i > max{4λ −1 ,4λ} 1−α1+α for all i ≥ 1.By the previous lemma, this does not affect the value of d ∗ (E+⌊λF⌋). With these assumptions,(I i +⌊λI j ⌋)∩(I k +⌊λI l ⌋) ≠ ∅ ⇐⇒ i = k and j,l < i or j = l and i,k < jand then(E +⌊λF⌋)∩(a i ,a i+1 ] =⊔(a+⌊λI j ⌋) , if i is odda∈I ij=2,4,...,i−1=⊔(I j +⌊λb⌋) , if i is evenb∈I ij=1,3,...,i−1In addition, (v) implies that, if i is odd, a,a ′ ∈ I i are distinct and j,k < i are even, then thegap between a+⌊λI j ⌋ and a ′ +⌊λI k ⌋ is at least |a−a ′ |/2; if i is even, b,b ′ ∈ I i are distinct andj,k < i are odd, then the gap between I j +⌊λb⌋ and I k +⌊λb ′ ⌋ is at least |b−b ′ |/4λ.By the above fractal-like structure, it is expected that E + ⌊λF⌋ has zero upper Banachdensity. Let us formally prove this. Consider an interval I = (M,N] ⊂ Z and, as usual, assumethat M +1,N ∈ E +⌊λF⌋, sayM +1 = a ′ +⌊λb ′ ⌋ ∈ I k +⌊λI l ⌋N = a+⌊λb⌋ ∈ I i +⌊λI j ⌋.

Chapter 4. Marstrand theorem for subsets of integers 49The largest index among i,j,k,l is either i or j. We may assume that i > j. In fact, thereverse case is symmetric, because E+⌊λF⌋ is basically ⌊λ(F +⌊λ −1 E⌋)⌋ and, with this interpretation,therolesofI i andI j areinterchanged. Inthissituation, wehavetwocases toconsider:• a = a ′ : an element r +⌊λs⌋ belongs to (E +⌊λF⌋)∩I iff r = a and s ∈ F ∩[b ′ ,b] and then|(E +⌊λF⌋)∩I||I|≤ |b−b′ +1| α|⌊λb⌋−⌊λb ′ ⌋| ∼ |b−b′ | αλ·|b−b ′ | = 1λ·|b−b ′ · (4.4.4)| 1−α• a > a ′ : for r + ⌊λs⌋ belong to (E + ⌊λF⌋) ∩ I, we necessarily have r ∈ E ∩ [a ′ ,a] ands ∈ I 2 ∪I 4 ∪···∪I i−1 and so|(E +⌊λF⌋)∩I||I|≤ |E ∩[a′ ,a]|·b i−1|a−a ′ |/2∼ |a−a′ | α ·b i−1|a−a ′ |≤ b i−1µ i1−α ≤ 1b i−1, (4.4.5)where in the last inequality we used (ii).Estimates (4.4.4) and (4.4.5) establish that E +⌊λF⌋ has zero upper Banach density.4.4.4 Another counterexample of Theorem 4.1.1WenowprovethatonecannotexpectthatthesetofparametersgivenbyTheorem4.1.1containsintegers. More specifically, we construct a regular set E ⊂ Z such that D(E) = D(E +λE) forevery λ ∈ Z. In particular, if D(E) ∈ (1/2,1), E +λE has zero upper Banach density and soevery parameter λ ∈ R for which d ∗ (E +⌊λE⌋) > 0 is not an integer.Fixed α ∈ (1/2,1) and c ∈ Z, consider the generalized IP-set associated to the sequencesObserve thatn−1∑i=1k n = c·2 n and d n =n−1∑k i ·d i ≤ c·i=12 i22α +i ≤ c·⌊2 n2 /2α ⌋ , n ≥ 1.(n−1) 22α +n−1∑j=12 j < c·2 (n−1)22α +n < d nfor large n. Also, by Lemma 4.3.6, this set has counting dimension α. Then, if E is thegeneralized IP-set associated tok n = 2 n and d n =⌊2 n2 /2α ⌋ ,thearithmeticsumE+λE isalsoageneralizedIP-set, associatedtothesequences ˜k n = (λ+1)·k nandd n , whichalsohascountingdimensionequaltoα. ByProposition4.4.3, E contains aregularsubset with dimension greater than 1/2. This subset gives the required counterexample.

Chapter 4. Marstrand theorem for subsets of integers 504.5 Proofs of the TheoremsLetE,F betwo regularcompatible subsetsof Z. Throughouttherestoftheproof, fixacompactinterval Λ of positive real numbers. Given distinct points z = (a,b) and z ′ = (a ′ ,b ′ ) of E ×F,letΛ z,z ′ = { λ ∈ Λ; a+⌊λb⌋ = a ′ +⌊λb ′ ⌋ } .Clearly, Λ z,z ′ is empty if b = b ′ . For b ≠ b ′ , it is possible to estimate its Lebesgue measure.Lemma 4.5.1. Let z = (a,b) and z ′ = (a ′ ,b ′ ) be distinct points of Z 2 . If Λ z,z ′ ≠ ∅, then(a) m(Λ z,z ′) |b−b ′ | −1 .(b) minΛ·|b−b ′ |−1 ≤ |a−a ′ | ≤ maxΛ·|b−b ′ |+1.Proof. Assume b > b ′ and let λ ∈ Λ z,z ′, say a+⌊λb⌋ = n = a ′ +⌊λb ′ ⌋. Then{n−a ≤ λb < n−a+1n−a ′ ≤ λb ′ < n−a ′ =⇒ a ′ −a−1 < λ(b−b ′ ) < a ′ −a+1+1and so( a ′ −a−1Λ z,z ′ ⊂b−b ′), a′ −a+1b−b ′ ,which proves (a). The second part also follows from the above inclusion, asanda ′ −a−1b−b ′ ≤ minΛ z,z ′ ≤ maxΛ =⇒ a ′ −a ≤ maxΛ·(b−b ′ )+1a ′ −a+1b−b ′≥ maxΛ z,z ′ ≥ minΛ =⇒ a ′ −a ≥ minΛ·(b−b ′ )−1.By item (b) of the above lemma, |a−a ′ | ∼ |b−b ′ | whenever Λ z,z ′ ≠ ∅. We point out that,although naive, Lemma 4.5.1 expresses the crucial property of transversality that makes theproof work, and all results related to Marstrand’s theorem use a similar idea in one way oranother.Let (I n ) n≥1 and (J n ) n≥1 be sequences of intervals satisfying the compatibility conditions ofDefinition 4.4.4. Associated to these intervals, consider, for each pair (n,λ) ∈ N×Λ, the setN n (λ) = { ((a,b),(a ′ ,b ′ )) ∈ ((E ∩I n )×(F ∩J n )) 2 ; a+⌊λb⌋ = a ′ +⌊λb ′ ⌋ }and, for each n ≥ 1, the integral∫∆ n = |N n (λ)|dm(λ).Λ

Chapter 4. Marstrand theorem for subsets of integers 51By a double counting, one has the equality∑∆ n = m(Λ z,z ′). (4.5.1)z,z ′ ∈(E∩I n)×(F×J n)Lemma 4.5.2. Let D(E) = α and D(F) = β as above.(a) If α+β < 1, then ∆ n |I n | α+β .(b) If α+β > 1, then ∆ n |I n | 2α+2β−1 .Proof. Using (4.5.1),and then∆ n ===∑z,z ′ ∈(E∩I n)×(F×J n)∑a∈E∩Inb∈F∩Jn∑a∈E∩Inb∈F∩Jn∑a∈E∩Inb∈F∩Jn |I n | α+β ·log|I n|∑s=1log|I n|∑s=1log|I n|∑s=1log|I n|∑s=1∑a ′ ∈E∩In|a−a ′ |∼e sm(Λ z,z ′)∑m(Λ z,z ′)b ′ ∈F∩Jn|b−b ′ |∼e se −s ·(e s ) α ·(e s ) β(e s ) α+β−1(e α+β−1) s∆ n |I n | α+β ·|I n | α+β−1 = |I n | 2α+2β−1 , if α+β > 1, |I n | α+β ·1 = |I n | α+β , if α+β < 1.Proof of Theorem 4.1.1. The proof is divided in three parts.Part 1. α+β < 1: fix ε > 0. By Lemma 4.5.2, the set of parameters λ ∈ Λ for which|N n (λ)| |I n| α+βhas Lebesgue measure at least m(Λ)−ε. We will prove thatε(4.5.2)|(E +⌊λF⌋)∩(I n +⌊λJ n ⌋)||I n +⌊λJ n ⌋| α+β ε (4.5.3)

Chapter 4. Marstrand theorem for subsets of integers 52whenever λ ∈ Λ satisfies (4.5.2). For each (m,n,λ) ∈ Z×Z×Λ, letThenands(m,n,λ) = |{(a,b) ∈ (E ∩I n )×(F ∩J n ); a+⌊λb⌋ = m}|.∑s(m,n,λ) = |E ∩I n |·|F ∩J n | ∼ |I n | α+β (4.5.4)m∈Z∑s(m,n,λ) 2 = |N n (λ)| |I n| α+β· (4.5.5)εm∈ZThe numerator in (4.5.3) is at least the cardinality of the set S(n,λ) = {m ∈ Z; s(m,n,λ) > 0},because (E + ⌊λF⌋) ∩ (I n + ⌊λJ n ⌋) contains S(n,λ). By the Cauchy-Schwarz inequality andrelations (4.5.4) and (4.5.5),and so, as |I n +⌊λJ n ⌋| ∼ |I n |,establishing (4.5.3).|S(n,λ)| ≥( ∑s(m,n,λ)m∈Z∑s(m,n,λ) 2m∈Z(|In | α+β) 2|I n | α+βε= ε·|I n | α+β) 2|(E +⌊λF⌋)∩(I n +⌊λJ n ⌋)||I n +⌊λJ n ⌋| α+β |S(n,λ)| ε,|I n | α+βFor each n ≥ 1, let G n ε = {λ ∈ Λ; (4.5.3) holds}. Then m(Λ\G n ε) ≤ ε, and the same holdsfor the setFor each λ ∈ G ε ,G ε = ⋂ n≥1∞⋃G l ε.l=nD α+β (E +⌊λF⌋) ≥ ε =⇒ D(E +⌊λF⌋) ≥ α+βand then, as the set G = ⋃ n≥1 G 1/n ⊂ Λ has Lebesgue measure m(Λ), Part 1 is completed.Part 2. α+β > 1: for a fixed ε > 0, Lemma 4.5.2 implies that the set of parameters λ ∈ Λ forwhich|N n (λ)| |I n| 2α+2β−1(4.5.6)ε

Chapter 4. Marstrand theorem for subsets of integers 53has Lebesgue measure at least m(Λ)−ε. In this case,|S(n,λ)| ≥( ∑s(m,n,λ)m∈Z∑s(m,n,λ) 2m∈Z(|In | α+β) 2|I n | 2α+2β−1ε= ε·|I n |) 2and then|(E +⌊λF⌋)∩(I n +⌊λJ n ⌋)||I n +⌊λJ n ⌋|The Borel-Cantelli argument is analogous to Part 1. |S(n,λ)||I n | ε.Part 3. α+β = 1: let n ≥ 1. Being regular, E has a regular subset E n ⊂ E, also compatible 5with F, such that D(E)−1/n < D(E n ) < D(E). Then1− 1 n < D(E n)+D(F) < 1and so, by Part 1, there is a full Lebesgue measure set Λ n such thatD(E n +⌊λF⌋) ≥ 1− 1 n , ∀λ ∈ Λ n.The set Λ = ⋂ n≥1 Λ n has full Lebesgue measure as well andD(E +⌊λF⌋) ≥ 1, ∀λ ∈ Λ.Proof of Theorem 4.1.3. We also divide it in parts.Part 1. ∑ ki=0 D(E i) ≤ 1: by Theorem 4.1.1,D(E 0 +⌊λ 1 E 1 ⌋) ≥ D(E 0 )+D(E 1 ) , m−a.e. λ 1 ∈ R.To each of these parameters, apply Proposition 4.4.3 to obtain a regular subset F λ1 ⊂ E 0 +⌊λ 1 E 1 ⌋ such that5 This may be assumed because of relation (4.4.1).D(F λ1 ) = D(E 0 )+D(E 1 ).

Chapter 4. Marstrand theorem for subsets of integers 54As E 2 is universal, another application of Theorem 4.1.1 guarantees thatD(F λ1 +⌊λ 2 E 2 ⌋) ≥ D(E 0 )+D(E 1 )+D(E 2 ), m−a.e. λ 2 ∈ Rand them, by Fubini’s theorem,D(E 0 +⌊λ 1 E 1 ⌋+⌊λ 2 E 2 ⌋) ≥ D(E 0 )+D(E 1 )+D(E 2 ), m 2 −a.e. (λ 1 ,λ 2 ) ∈ R 2 .Iterating the above arguments, it follows thatD(E 0 +⌊λ 1 E 1 ⌋+···+⌊λ k E k ⌋) ≥ D(E 0 )+···+D(E k ), m k −a.e. (λ 1 ,...,λ k ) ∈ R k .Part 2. ∑ ki=0 D(E i) > 1: without loss of generality, we can assumeD(E 0 )+···+D(E k−1 ) ≤ 1 < D(E 0 )+···+D(E k−1 )+D(E k ).By Part 1,D(E 0 +⌊λ 1 E 1 ⌋+···+⌊λ k−1 E k−1 ⌋) ≥ D(E 0 )+···+D(E k−1 )for m k−1 −a.e. (λ 1 ,...,λ k−1 ) ∈ R k−1 . To each of these (k−1)-tuples, consider a regular subsetF (λ1 ,...,λ k−1 ) of E 0 +···+⌊λ k−1 E k−1 ⌋ such thatD ( F (λ1 ,...,λ k−1 ))= D(E0 )+···+D(E k−1 ).Finally, because D(F (λ1 ,...,λ k−1 ))+D(E k ) > 1, Theorem 4.1.1 guarantees thatd ∗( F (λ1 ,...,λ k−1 ) +⌊λ k E k ⌋ ) > 0 , m−a.e. λ k ∈ R,which, after another application of Fubini’s theorem, concludes the proof.4.6 Concluding remarksWe think there is a more specific way of defining the counting dimension that encodes theconditions of regularity and compatibility. A natural candidate would be a prototype of aHausdorff dimension, where one looks to all covers, properly renormalized in the unit interval,and takes a liminf. An alternative definition has appeared in [35]. It would be a naturalprogram to prove Marstrand type results in this context.Another interesting question is to consider arithmetic sums E+λF, where λ ∈ Z. These aregenuine arithmetic sums and, as we saw in Subsection 4.4.4, their dimension may not increase.We think very strong conditions on the sets E,F are needed to prove analogous results aboutE +λF for λ ∈ Z.

Chapter 4. Marstrand theorem for subsets of integers 55We also think the results obtained in this chapter work to subsets of Z k . Given E ⊂ Z k , theupper Banach density of E is equal tod ∗ |E ∩(I 1 ×···×I k )|(E) = limsup,|I 1 |,...,|I k |→∞ |I 1 ×···×I k |where I 1 ,...,I k run over all intervals of Z, the counting dimension of E islog|E ∩(I 1 ×···×I k )|D(E) = limsup,|I 1 |,...,|I k |→∞ log|I 1 ×···×I k |where I 1 ,...,I k run over all intervals of Z and, for α ≥ 0, the counting α-measure of E is|E ∩(I 1 ×···×I k )|H α (E) = limsup|I 1 |,...,|I k |→∞ |I 1 ×···×I k | α ,whereI 1 ,...,I k runover all intervals of Z. These quantities satisfy similar properties to those inSubsection 4.2.2. The notion of regularity is defined in an analogous manner. For compatibility,we have to take into account the geometry of Z k . Two regular subsets E,F ⊂ Z k are compatibleif there exist sequences of rectangles R n = I1 n ×··· ×In k and S n = J1 n ×··· ×Jn k, n ≥ 1, suchthat(i) |Ii n| ∼ |Jn i | for every i = 1,2,...,k, and(ii) |E ∩R n | |R n | D(E) and |F ∩S n | |S n | D(F) .We think the theory of this chapter may be extended to prove that if E,F ⊂ Z k are two regularcompatible subsets, thenD(E +⌊λF⌋) ≥ min{1,D(E) +D(F)}for Lebesgue almost every λ ∈ R and, if in addition D(E) + D(F) > 1, then E + ⌊λF⌋ haspositive upper Banach density for Lebesgue almost every λ ∈ R.

Chapter 4. Marstrand theorem for subsets of integers 56

Chapter 5Z d -actions with prescribedtopological and ergodic propertiesWe extend constructions of Hahn-Katznelson [17] and Pavlov [40] to Z d -actions on symbolicdynamical spaces with prescribed topological and ergodic properties. More specifically, wedescribe a method to build Z d -actions which are (totally) minimal, (totally) strictly ergodicand have positive topological entropy.5.1 IntroductionErgodic theory studies statistical and recurrence properties of measurable transformations Tacting in a probability space (X,B,µ), where µ is a measure invariant by T, that is, µ(T −1 A) =µ(A), forallA ∈ B. Itinvestigates awideclassofnotions, suchasergodicity, mixingandentropy.These properties, in some way, give qualitative and quantitative aspects of the randomness of T.For example, ergodicity means that T is indecomposable in the metric sense with respect to µand entropy is a concept that counts the exponential growth rate for the number of statisticallysignificant distinguishable orbit segments.In most cases, the object of study has topological structures: X is a compact metric space,B is the Borel σ-algebra of X, µ is a Borel measure probability and T is a homeomorphism of X.In this case, concepts such as minimality and topological mixing give topological aspects of therandomness of T. For example, minimality means that T is indecomposable in the topologicalsense, that is, the orbit of every point is dense in X.A natural question arises: how do ergodic and topological concepts relate to each other?How do ergodic properties forbid topological phenomena and vice-versa? Are metric and topologicalindecomposability equivalent? This last question was answered negatively in [15] via57

Chapter 5. Z d -actions with prescribed topological and ergodic properties 58the construction of a minimal diffeomorphism of the torus T 2 which preserves area but is notergodic.AnotherquestionwasraisedbyW.Parry: supposeT hasauniqueBorelprobabilityinvariantmeasure and that (X,T) is a minimal transformation. Can (X,T) have positive entropy? Thedifficulty in answering this at the time was the scarcity of a wide class of minimal and uniquelyergodictransformations. Thiswas solved affirmatively in [17], whereF. HahnandY. Katznelsondeveloped an inductive method of constructing symbolic dynamical systems with the requiredproperties. The principal idea of the paper was the weak law of large numbers.Later, works of Jewett and Krieger (see [42]) proved that every ergodic measure-preservingsystem (X,B,µ,T) is metrically isomorphic to a minimal and uniquely ergodic homeomorphismon a Cantor set and this gives many examples to Parry’s question: if an ergodic system(X,B,µ,T) has positive metric entropy and Φ : (X,B,µ,T) → (Y,C,ν,S) is the metric isomorphismobtained by Jewett-Krieger’s theorem, then (Y,S) has positive topological entropy, bythe variational principle.It is worth mentioning that the situation is quite different in smooth ergodic theory, oncesome regularity on the transformation is assumed. A. Katok showed in [21] that every C 1+αdiffeomorphism of a compact surface can not be minimal and simultaneously have positivetopological entropy. More specifically, he proved that the topological entropy can be written interms of the exponential growth of periodic points of a fixed order.Suppose that T is a mesure-preserving transformation on the probability space (X,B,µ)and f : X → R is a measurable function. A successful area in ergodic theory deals withthe convergence of averages n −1·∑nk=1 f ( T k x ) , x ∈ X, when n converges to infinity. The wellknown Birkhoff’s Theorem states that such limit exists for almost every x ∈ X whenever f is anL 1 -function. Several resultshave been(and still are being) proved when, instead of {1,2,...,n},average is made along other sequences of natural numbers. A remarkable result on this directionwasgivenbyJ.Bourgain[6], whereheprovedthatifp(x)isapolynomialwithintegercoefficientsand f is an L p -function, for some p > 1, then the averages n −1 ·∑nk=1 f ( T p(k) x ) converge foralmost every x ∈ X. In other words, convergence fails to hold for a negligible set with respectto the measure µ. We mention this result is not true for L 1 -functions, as recently proved by Z.Buczolich and D. Mauldin [7].In [3], V. Bergelson asked if this set is also negligible from the topological point of view. Itturnedout, by aresult of R. Pavlov [40], that this is not true. Heproved that, for every sequence(p n ) n≥1 ⊂ Zofzeroupper-Banachdensity, thereexistatotally minimal, totally uniquelyergodicand topologically mixing transformation (X,T) and a continuous function f : X → R such thatn −1 ·∑nk=1 f (Tp kx) fails to converge for a residual set of x ∈ X.Suppose now that (X,T) is totally minimal, that is, (X,T n ) is minimal for every positive

Chapter 5. Z d -actions with prescribed topological and ergodic properties 59integer n. Pavlov alsoprovedthat, foreverysequence(p n ) n≥1 ⊂ Zofzeroupper-Banachdensity,there exists a totally minimal, totally uniquely ergodic and topologically mixing continuoustransformation (X,T) such that x ∉ {T pn x; n ≥ 1} for an uncountable number of x ∈ X.In this work, we extend the results of Hahn-Katznelson and Pavlov, giving a method of constructing(totally) minimal and (totally) uniquely ergodic Z d -actions with positive topologicalentropy. We carry out our program by constructing closed shift invariant subsets of a sequencespace. More specifically, we build a sequence of finite configurations (C k ) k≥1 of {0,1} Zd , C k+1being essentially formed by the concatenation of elements in C k such that each of them occursstatistically well-behaved in each element of C k+1 , and consider the set of limits of shiftedC k -configurations as k → ∞. The main results areTheorem 5.1.1. There exist totally strictly ergodic Z d -actions (X,B,µ,T) with arbitrarily largepositive topological entropy.We should mention that this result is not new, because Jewett-Krieger’s Theorem is true forZ d -actions [46]. This formulation emphasizes to the reader that the constructions, which maybe used in other settings, have the additional advantage of controlling the topological entropy.Theorem 5.1.2. Given a set P ⊂ Z d of zero upper-Banach density, there exist a totally strictlyergodic Z d -action (X,B,µ,T) and a continuous function f : X → R such that the ergodicaverages1|P ∩(−n,n) d |∑g∈P∩(−n,n) d f (T g x)fail to converge for a residual set of x ∈ X. In addition, (X,B,µ,T) can have arbitrarily largetopological entropy.The above theorem has a special interest when P is an arithmetic set for which classicalergodic theory and Fourier analysis have established almost-sure convergence. This is the case(also proved in [6]) whenP = {(p 1 (n),...,p d (n)); n ∈ Z},where p 1 ,...,p d are polynomials with integer coefficients: for any f ∈ L p , p > 1, the limit1n∑ ( )lim f T (p 1(k),...,p d (k)) xn→∞ nk=1exists almost-surely. Note that P has zero upper-Banach density whenever one of the polynomialshas degree greater than 1.Theorem 5.1.3. Given a set P ⊂ Z d of zero upper-Banach density, there exists a totally strictlyergodic Z d -action (X,B,µ,T) and an uncountable set X 0 ⊂ X for which x ∉ {T p x; p ∈ P}, forevery x ∈ X 0 . In addition, (X,B,µ,T) can have arbitrarily large topological entropy.

Chapter 5. Z d -actions with prescribed topological and ergodic properties 60Yet in the arithmetic setup, Theorem 5.1.3 is the best topological result one can expect.Indeed, Bergelson and Leibman proved in [4] that if T is a minimal Z d -action, then there is aresidual set Y ⊂ X for which x ∈ {T (p 1(n),...,p d (n)) x; n ∈ Z}, for every x ∈ Y.5.2 PreliminariesWe begin with some notation. Consider a metric space X, B its Borel σ-algebra and a G groupwith identity e. Throughout this work, G will denote Z d , d > 1, or one of its subgroups.5.2.1 Group actionsDefinition 5.2.1. A G-action on X is a measurable transformation T : G×X → X, denotedby (X,T), such that(i) T(g 1 ,T(g 2 ,x)) = T(g 1 g 2 ,x), for every g 1 ,g 2 ∈ G and x ∈ X.(ii) T(e,x) = x, for every x ∈ X.In other words, for each g ∈ G, the restrictionT g : X −→ Xx ↦−→ T(g,x)is a bimeasurable transformation on X such that T g 1g 2= T g 1T g 2, for every g 1 ,g 2 ∈ G. WhenG is abelian, (T g ) g∈G forms a commutative group of bimeasurable transformations on X. Foreach x ∈ X, the orbit of X with respect to T is the setO T (x) = . {T g x; g ∈ G}.If F is a subgroup of G, the restriction T| F : F ×X → X is clearly a F-action on X.Definition 5.2.2. We say that (X,T) is minimal if O T (x) is dense in X, for every x ∈ X, andtotally minimal if O T|F (x) is dense in X, for every x ∈ X and every subgroup F < G of finiteindex.Remind that the index of a subgroup F, denoted by (G : F), is the number of cosets of F inG. The above definition extends the notion of total minimality of Z-actions. In fact, a Z-action(X,T) is totally minimal if and only if T n : X → X is a minimal transformation, for everyn ∈ Z.Consider the set M(X) of all Borel probability measures in X. A probability µ ∈ M(X) isinvariant under T or simply T-invariant ifµ(T g A) = µ(A), ∀g ∈ G, ∀A ∈ B.

Chapter 5. Z d -actions with prescribed topological and ergodic properties 61Let M T (X) ⊂ M(X) denote the set of all T-invariant probability measures. Such set isnon-empty whenever G is amenable, by a Krylov-Bogolubov argument applied to any Fφlnersequence 1 of G.Definition 5.2.3. A G measure-preserving system or simply G-mps is a quadruple (X,B,µ,T),where T is a G-action on X and µ ∈ M T (X).We say that A ∈ B is T-invariant if T g A = A, for all g ∈ G.Definition 5.2.4. The G-mps (X,B,µ,T) is ergodic if it has only trivial invariant sets, that is,if µ(A) = 0 or 1 whenever A is a measurable set invariant under T.Definition 5.2.5. The G-action (X,T) is uniquely ergodic if M T (X) is unitary, and totallyuniquely ergodic if, for every subgroup F < G of finite index, the restricted F-action (X,T| F )is uniquely ergodic.Definition 5.2.6. We say that (X,T) is strictly ergodic if it is minimal and uniquely ergodic,and totally strictly ergodic if, for every subgroup F < G of finite index, the restricted F-action(X,T| F ) is strictly ergodic.The result below was proved in [47] and states the pointwise ergodic theorem for Z d -actions.Theorem 5.2.7. Let (X,B,µ,T) be a Z d -mps. Then, for every f ∈ L 1 (µ), there is a T-invariant function ˜f ∈ L 1 (µ) such thatlimn→∞1 ∑n d f (T g x) = ˜f(x)g∈[0,n) dfor µ-almost every x ∈ X. In particular, if the action is ergodic, ˜f is constant and equal to∫fdµ.Above, [0,n) denotestheset{0,1,...,n−1}, [0,n) d thed-dimensional cube[0,n)×···×[0,n)of Z d and by a T-invariant function we mean that f ◦ T g = f, for every g ∈ G. These averagesallow the characterization of unique ergodicity. Let C(X) denote the space of continuousfunctions from X to R.Proposition 5.2.8. Let (X,T) be a Z d -action on the compact metric space X. The followingitems are equivalent.(a) (X,T) is uniquely ergodic.1 (A n) n≥1 is a Fφlner sequence of G if lim n→∞|A n∆gA n|/|A n| = 0 for every g ∈ G.

Chapter 5. Z d -actions with prescribed topological and ergodic properties 62(b) For every f ∈ C(X) and x ∈ X, the limitexists and is independent of x.limn→∞(c) For every f ∈ C(X), the sequence of functions1 ∑n d f (T g x)g∈[0,n) df n = 1 n d ∑converges uniformly in X to a constant function.g∈[0,n) d f ◦T gProof. The implications (c)⇒(b)⇒(a) are obvious. It remains to prove (a)⇒(c). Let M T (X) ={µ}. We’ll show that f n converges uniformly to ˜f = ∫ fdµ. By contradiction, suppose this isnot the case for some f ∈ C(X). This means that there exist ε > 0, n i → ∞ and x i ∈ X suchthat ∣ ∣∣∣f ni (x i )−∫fdµ∣ ≥ ε.For each i, let ν i ∈ M(X) be the probability measure associated to the linear functionalΘ i : C(X) → R defined byΘ i (ϕ) = 1n id∑g∈[0,n i ) d ϕ(T g x i ), ϕ ∈ C(X).Restricting to a subsequence, if necessary, we assume that ν i → ν in the weak-star topology.Because the cubes A i = [0,n i ) d form a Fφlner sequence in Z d , ν ∈ M T (X). In fact, for eachh ∈ Z d ,(∣∣∫ϕ◦T h) ∫dν −ϕdν∣ = lim1i→∞ n d i∣∑g∈A i +hϕ(T g x i )− ∑ ϕ(T g x i )g∈A i∣#A i ∆(A i +h)≤ max|ϕ(x)|· limx∈X i→∞ #A i= 0.But∣ ∣∣∣∫∫fdν −∫fdµ∣ = lim∣i→∞∫fdν i −∫fdµ∣ = lim∣ f n i(x i )−i→∞fdµ∣ ≥ εand so ν ≠ µ, contradicting the unique ergodicity of (X,T).

Chapter 5. Z d -actions with prescribed topological and ergodic properties 635.2.2 Subgroups of Z dLet F be the set of all subgroups of Z d of finite index. This set is countable, because eachelement of F is generated by d linearly independent vectors of Z d . Consider, then, a subset(F k ) k∈N of F such that, for each F ∈ F, there exists k 0 > 0 such that F k < F, for everyk ≥ k 0 . For this, just consider an enumeration of F and define F k as the intersection of thefirst k elements. Such intersections belong to F because(Z d : F ∩F ′ ) ≤ (Z d : F)·(Z d : F ′ ), ∀F,F ′ < Z d .Restricting them, if necessary, we assume that F k = m k · Z d , where (m k ) k≥1 is an increasingsequence of positive integers. Observe that (Z d : F k ) = m d k . Such sequence will be fixedthroughout the rest of the chapter.Definition 5.2.9. Given asubgroupF < Z d , wesaythattwoelements g 1 ,g 2 ∈ Z d arecongruentmodulo F if g 1 −g 2 ∈ F and denote it by g 1 ≡ F g 2 . The set ¯F ⊂ Z d is a complete residue setmodulo F if, for every g ∈ Z d , there exists a unique h ∈ ¯F such that g ≡ F h.Every complete residue set modulo F is canonically identified to the quocient Z d /F and hasexactly (Z d : F) elements.5.2.3 Symbolic spacesLetC beafinitealphabet andconsidertheset Ω(C) = C Zd of all functionsx : Z d → C. We endowC withthediscrete topology andΩ(C) withtheproducttopology. By Tychonoff’s theorem, Ω(C)is a compact metric space. We are not interested in a particular metric in Ω(C). Instead, weconsider a basis of topology B 0 to be defined below.Consider the family R of all finite d-dimensional cubes A = [r 1 ,r 1 +n)×··· ×[r d ,r d +n)of Z d , n ≥ 0. We say that A has length n and is centered at g = (r 1 ,...,r d ) ∈ Z d .Definition 5.2.10. A configuration or pattern is a pair b A = (A,b), where A ∈ R and b is afunction from A to C. We say that b A is supported in A with encoding function b.Let Ω A (C) denote the space of configurations supported in A and Ω ∗ (C) the space of allconfigurations in Z d :Ω ∗ (C) = . {b A ; b A is a configuration}.Given A ∈ R, consider the map Π A : Ω(C) → Ω A (C) defined by the restrictionΠ A (x) : A −→ Cg ↦−→ x(g)In particular, Π {g} (x) = x(g). We use the simpler notation x| A to denote Π A (x).

Chapter 5. Z d -actions with prescribed topological and ergodic properties 64Definition 5.2.11. If A ∈ R is centered at g, we say that x| A is a configuration of x centeredat g or that x| A occurs in x centered at g.For A 1 ,A 2 ∈ R such that A 1 ⊂ A 2 , let π A 2A 1: Ω A2 → Ω A1 be the restrictionπ A 2A 1(b) : A 1 −→ Cg ↦−→ b(g)As above, when there is no ambiguity, we denote π A 2A 1(b) simply by b| A1 . It is clear that thediagram below commutes.Π A2Ω(C) Ω A2 (C)Π A1π A 2A 1Ω A1 (C)These maps will help us to control the patterns to appear in the constructions of Section 5.3.By a cylinder in Ω(C) we mean the set of elements of Ω(C) with some fixed configuration.More specifically, given b A ∈ Ω ∗ (C), the cylinder generated by b A is the setCyl(b A ) . = {x ∈ Ω(C); x| A = b A }.The family B 0 := {Cyl(b A )|b A ∈ Ω ∗ (C)} forms a clopen set of cylinders generating B. Hencethe set C 0 = {χ B ; B ∈ B 0 } of cylinder characteristic functions generates a dense subspace inC(Ω(C)). Let µ be the probability measure defined byµ(Cyl(b A )) = |C| −|A| , ∀b A ∈ Ω ∗ (C),and extended to B by Caratheódory’s Theorem. Above, |·| denotes the number of elements ofa set.Consider the Z d -action T : Z d ×Ω(C) → Ω(C) defined byT g (x) = (x(g +h)) h∈Z d ,also called the shift action. Given B = Cyl(b A ) and g ∈ Z d , let B + g denote the cylinderassociated to b A+g = (˜b,A + g), where ˜b : A + g → {0,1} is defined by ˜b(h) = b(h − g),∀h ∈ A+g. With this notation,χ B ◦T g = χ B+g . (5.2.1)

Chapter 5. Z d -actions with prescribed topological and ergodic properties 65In fact,⇐⇒⇐⇒⇐⇒⇐⇒χ B (T g x) = 1T g x ∈ Bx(g +h) = b(h), ∀h ∈ Ax(h) = ˜b(h), ∀h ∈ A+gx ∈ B +g.Definition 5.2.12. A subshift of (Ω(C),T) is a Z d -action (X,T), where X is a closed subset ofΩ(C) invariant under T.5.2.4 Topological entropyFor each subset X of Ω(C) and A ∈ R, letΩ A (C,X) = {x| A ; x ∈ X}denote the set of configurations supported in A which occur in elements of X and Ω ∗ (C,X) thespace of all configurations in Z d occuring in elements of X,Ω ∗ (C,X) = ⋃Ω A (C,X).A∈RDefinition 5.2.13. The topological entropy of the subshift (X,T) is the limitlog|Ω [0,n) d(C,X)|h(X,T) = limn→∞ n d , (5.2.2)which always exists and is equal to inf n∈N1n d ·log|Ω [0,n) d(C,X)|.5.2.5 Frequencies and unique ergodicityDefinition 5.2.14. Given configurations b A1 ∈ Ω A1 (C) and b A2 ∈ Ω A2 (C), the set of ocurrencesof b A1 in b A2 isS(b A1 ,b A2 ) . = {g ∈ Z d ; A 1 +g ⊂ A 2 and π A 2A 1 +g (b A 2) = b A1 +g}.The frequency of b A1 in b A2 is defined asfr(b A1 ,b A2 ) . = |S(b A 1,b A2 )||A 2 |Given F ∈ F and h ∈ Z d , the set of ocurrences of b A1 in b A2 centered at h modulo F isS(b A1 ,b A2 ,h,F) . = {g ∈ S(b A1 ,b A2 ); A 1 +g is centered at a vertex ≡ F h}·

Chapter 5. Z d -actions with prescribed topological and ergodic properties 66and the frequency of b A1 in b A2 centered at h modulo F is the quocientfr(b A1 ,b A2 ,h,F) . = |S(b A 1,b A2 ,h,F)||A 2 |Observe that if ¯F ⊂ Z d is a complete residue set modulo F, thenfr(b A1 ,b A2 ) = ∑ g∈¯Ffr(b A1 ,b A2 ,g,F).·To our purposes, we rewrite Proposition 5.2.8 in a different manner.Proposition 5.2.15. A subshift (X,T) is uniquely ergodic if and only if, for every b A ∈ Ω ∗ (C)and x ∈ X,exists and is independent of x.fr(b A ,x) = . ( )lim fr b A ,x|n→∞ [0,n) dProof. By approximation, condition (b) of Proposition 5.2.8 holds for C(X) if and only if itholds for C 0 = {χ B ; B ∈ B 0 }. If f = χ Cyl(bA ), (5.2.1) implies thatlim f 1 ∑n(x) = limn→∞ n→∞ n d f (T g x)g∈[0,n) d1 ∑= limn→∞ n d χ Cyl(bA+g) (x)g∈[0,n) d1 ∑= limn→∞ n d χ Cyl(bA+g) (x)g∈[0,n) dA+g⊂[0,n)(d )= lim fr b A ,x|n→∞ [0,n) d= fr(b A ,x),where in the third equality we used that, for a fixed A ∈ R,|{g ∈ [0,n) d ; A+g ⊄ [0,n) d }|limn→∞ n d = 0.Corollary 5.2.16. A subshift (X,T) is totally uniquely ergodic if and only if, for every b A ∈Ω ∗ (C), x ∈ X and F ∈ F,exists and is independent of x.fr(b A ,x,F) = . ( )lim fr b A ,x|n→∞ [0,n) d,0,FSo, uniqueergodicity is all aboutconstant frequencies. We’ll obtain thisviatheLaw ofLargeNumbers, equidistributing ocurrences of configurations along residue classes of subgroups.

Chapter 5. Z d -actions with prescribed topological and ergodic properties 675.2.6 Law of Large NumbersIntuitively, if A is a subset of Z d , each letter of C appears in x| A with frequency approximately1/|C|, for almost every x ∈ Ω(C). This is what the Law of Large Number says. For our purposes,we state this result in a slightly different way. Let (X,B,µ) be a probability space and A ⊂ Z dinfinite. For each g ∈ A, let X g : X → R be a random variable.Theorem 5.2.17. (Law of Large Numbers) If (X g ) g∈A is a family of independent and identicallydistributed random variables such that E[X g ] = m, for every g ∈ A, then the sequence ( X n)n≥1defined byX n =∑g∈A∩[0,n) d X g|A∩[0,n) d |converges in probability to m, that is, for any ε > 0,lim µ(∣ ∣ Xn −m ∣ ) < ε = 1.n→∞Consider the probability measure space (X,B,µ) defined in Subsection 5.2.3. Fixed w ∈ C,let X g : Ω(C) → R be defined asX g (x) = 1 , if x(g) = w= 0 , if x(g) ≠ w.(5.2.3)It is clear that (X g ) g∈Z d are independent, identically distributed and satisfy∫E[X g ] = X g (x)dµ(x) = 1|C| , ∀g ∈ Zd .In addition,which implies theX n (x) =X∑g∈[0,n) d X g(x)n d =(∣ ∣∣∣S w,x| [0,n) d)∣Corollary 5.2.18. Let w ∈ C, g ∈ Z d , F ∈ F and ε > 0.(a) The number of elements b ∈ Ω [0,n) d(C) such that∣ fr(w,b)− 1|C| ∣ < εis asymptotic to |C| nd as n → ∞.(b) The number of elements b ∈ Ω [0,n) d(C) such that∣ fr(w,b,g,F)− 1|C|·(Z d : F) ∣ < εis asymptotic to |C| nd as n → ∞.n d( )= fr w,x| [0,n) d ,

Chapter 5. Z d -actions with prescribed topological and ergodic properties 68(c) The number of elements b ∈ Ω [0,n) d(C) such that∣ fr(w,b,g,F)− 1|C|·(Z d : F) ∣ < εfor every w ∈ C and g ∈ Z d is asymptotic to |C| nd as n → ∞.Proof. (a) The required number is equal to(|C| nd ·µ {x ∈ Ω(C);∣∣fr( ∣ )w,x| [0,n) d)−|C| −1 ∣∣ < ε}and is asymptotic to |C| nd , as the above µ-measure converges to 1.(b) Take A = F +g and (X h ) h∈A as in (5.2.3). For any x ∈ Ω(C),(∣ ∣∣∣S w,x| [0,n) d,g,F)∣X n (x) =|A∩[0,n) d |( )= fr w,x| [0,n) d,g,F ·(Z d : F)+o(1),because ∣ ∣ A∩[0,n)d is asymptotic to n d /(Z d : F). This implies that for n large)∣ ∣(w,x| fr 1∣∣∣[0,n) d,g,F −|C|·(Z d : F) ∣ < ε ⇐⇒ X n (x)− 1|C| ∣ < ε·(Zd : F)and then Theorem 5.2.17 guarantees the conclusion.(c) As the events are independent, this follows from (b).5.3 Main ConstructionsLet C = {0,1}. In this section, we construct subshifts (X,T) with topological and ergodic prescribedproperties.To this matter, we buildasequenceof finitenon-empty sets of configurationsC k ⊂ Ω Ak (C), k ≥ 1, such that:(i) A k = [0,n k ) d , where (n k ) k≥1 is an increasing sequence of positive integers.(ii) n 1 = 1 and C 1 = Ω A1 (C) ∼ = {0,1}.(iii) C k is the concatenation of elements of C k−1 , possibly with the insertion of few additionalblocks of zeroes and ones.Given such sequence (C k ) k≥1 , we consider X ⊂ Ω(C) as the set of limits of shifted C k -patternsas k → ∞, that is, x ∈ X if there exist sequences (w k ) k≥1 , w k ∈ C k , and (g k ) k≥1 ⊂ Z d such thatx = limk→∞ Tg kw k .

Chapter 5. Z d -actions with prescribed topological and ergodic properties 69The above limit has an abuse of notation, because T acts in Ω(C) and w k ∉ Ω(C). Formallyspeaking, this means that, for each g ∈ Z d , there exists k 0 ≥ 1 such thatx(g) = w k (g +g k ), ∀k ≥ k 0 .By definition, X is invariant under T and, for any k, every x ∈ X is an infinite concatenationof elements of C k and additional blocks of zeroes and ones.If C k ⊂ Ω Ak ({0,1}) and A ∈ R, Ω A (C k ) is identified in a natural way to a subset ofΩ nk A({0,1}). In some situations, to distinguish this association, we use small letters for Ω A (C k )and capital letters for Ω nk A({0,1}) 2 . In this situation, if w ∈ Ω A (C k ) and g ∈ A, the patternw(g) ∈ C k occurs in W ∈ Ω nk A({0,1}) centered at n k g. In other words, if w k ∈ C k , thenS(w k ,W,n k g,F) = n k ·S(w k ,w,g,F). (5.3.1)In each of the next subsections, (C k ) k≥1 is constructed with specific combinatorial andstatistical properties.5.3.1 MinimalityThe action (X,T) is minimal if and only if, for each x,y ∈ X, every configuration of x is also aconfiguration of y. For this, suppose C k ⊂ Ω Ak ({0,1}) is defined and non-empty.By the Law of Large Numbers, if l k is large, every element of C k occurs in almost everyelement of Ω [0,lk ) d(C k) (in fact, by Corollary 5.2.18, each of them occurs approximately withfrequency 1/|C k | > 0). Take any subset C k+1 of Ω [0,lk ) d(C k) with this property and consider itas a subset of Ω [0,nk+1 ) d({0,1}), where n k+1 = l k n k .Let us prove that (X,T) is minimal. Consider x,y ∈ X and x| A a finite configuration of x.For large k, x| A is a subconfiguration of some w k ∈ C k . As y is formed by the concatenation ofelements of C k+1 , every element of C k is a configuration of y. In particular, w k (and then x| A )is a configuration of y.5.3.2 Total minimalityThe action (X,T) is totally minimal if and only if, for each x,y ∈ X and F ∈ F, everyconfiguration x| A of x centered 3 at 0 also occurs in y centered at some g ∈ F. To guarantee thisfor every F ∈ F, we inductively control the ocurrence of subconfigurations centered in finitelymany subgroups of Z d .2 For example, w ∈ Ω A(C k ) and W ∈ Ω nk A({0,1}) denote the “same” element.3 Because of the T-invariance of X, we can suppose that x| A is centered in 0 ∈ Z d . In fact, instead of x,y, weconsider T g x,T g y.

Chapter 5. Z d -actions with prescribed topological and ergodic properties 70Consider the sequence (F k ) ⊂ F defined in Subsection 5.2.2. By induction, suppose C k ⊂Ω Ak ({0,1}) is non-empty satisfying (i), (ii), (iii) and the additional assumption(iv) gcd(n k ,m k ) = 1 (observe that this holds for k = 1).Take l k large and ˜C k+1 ⊂ Ω [0,lk m k+1 ) d(C k) non-empty such that(v) S(w k ,w| [0,lk m k+1 −1) d,g,F k) ≠ ∅, for every triple (w k ,w,g) ∈ C k × ˜C k+1 ×Z d .Considering w| [0,lk m k+1 −1) d as an element of Ω [0,l k m k+1 n k −n k ) d({0,1}), (5.3.1) implies thatS(w k ,W| [0,lk m k+1 n k −n k ) d,n kg,F k ) ≠ ∅, ∀(w k ,w,g) ∈ C k × ˜C k+1 ×Z d .As gcd(n k ,m k ) = 1, the set n k Z d runs over all residue classes modulo F k and so (the restrictionto [0,l k m k+1 n k −n k ) d of) every element of ˜C k+1 contains every element of C k centered at everyresidue class modulo F k .Obviously, C k+1 must not be equal to ˜C k+1 , because m k+1 divides l k m k+1 n k . Instead, wetake n k+1 = l k m k+1 n k + 1 and insert positions B i , i = 1,2,...,d, next to faces of the cube[0,l k m k+1 n k ) d . These are given byB i = {(r 1 ,...,r d ) ∈ A k+1 ; r i = l k m k+1 n k −n k }.There is a natural surjection Φ : Ω Ak+1 ({0,1}) → Ω [0,nk+1 −1) d({0,1}) obtained removing thepositions B 1 ,...,B d . More specifically, ifδ(r) = 0, if r < l k m k+1 n k −n k ,= 1, otherwiseand∆(r 1 ,...,r d ) = (δ(r 1 ),...,δ(r d )), (5.3.2)the map Φ is given byΦ(W)(g) = W (g +∆(g)), ∀(r 1 ,...,r d ) ∈ [0,n k+1 −1) d .We conclude the induction step taking C k+1 = Φ −1 (˜C k+1 ).

Chapter 5. Z d -actions with prescribed topological and ergodic properties 717 8 94 5 6Φ7 8 94 5 61 2 31 2 3Φ(w k+1 ) ∈ ˜C k+1 w k+1 ∈ C k+1Figure: example of Φ when d = 2.By definition, w k+1 and Φ(w k+1 ) coincide in [0,n k+1 −n k −1) d , for every w k+1 ∈ C k+1 . Thisimplies that every element of C k appears in every element of C k+1 centered at every residue classmodulo F k .Let us prove that (X,T) is totally minimal. Fix elements x,y ∈ X, a subgroup F ∈ F anda pattern x| A of x centered in 0 ∈ Z d . By the definition of X, x| A is a subconfiguration ofsome w k ∈ C k , for k large enough such that F k < F. As y is built concatenating elements ofC k+1 , w k occurs in y centered in every residue class modulo F and the same happens to x| A . Inparticular, x| A occurs in y centered in some g ∈ F, which is exactly the required condition.5.3.3 Total strict ergodicityIn addition to the occurrence of configurations in every residue class of subgroups of Z d , wealso control their frequency. Consider a sequence (d k ) k≥1 of positive real numbers such that∑k≥1 d k < ∞. Assume that C 1 ,...,C k−1 ,C k are non-empty sets satisfying (i), (ii), (iii), (iv)and(vi) For every (w k−1 ,w k ,g) ∈ C k−1 ×C k ×Z d ,(1−dk−1fr(w k−1 ,w k ,g,F k−1 ) ∈m k−1d ·|C k−1 |,)1+d k−1·m k−1d ·|C k−1 |Before going to the inductive step, let us make an observation. Condition (vi) also controls thefrequency on subgroups F such that F k−1 < F. In fact, if ¯F k−1 is a complete residue set moduloF k−1 ,fr(w k−1 ,w k ,g,F) = ∑h∈ ¯F k−1h≡ F gfr(w k−1 ,w k ,h,F k−1 ) (5.3.3)and, as ∣ {h ∈ ¯Fk−1 ; h ≡ F g} ∣ = (F : Fk−1 ),(1−d k−1fr(w k−1 ,w k ,g,F) ∈(Z d : F)·|C k−1 |,)1+d k−1(Z d · (5.3.4): F)·|C k−1 |

Chapter 5. Z d -actions with prescribed topological and ergodic properties 72Weproceedthesamewayasintheprevioussubsection: takel k largeand ˜C k+1 ⊂ Ω [0,lk m k+1 ) d(C k)non-empty such thatfr(w k , ˜w k+1 ,g,F k ) ∈( 1−dkm kd ·|C k |,)1+d km kd ·|C k |(5.3.5)for every (w k , ˜w k+1 ,g) ∈ C k × ˜C k+1 ×Z d . Note that the non-emptyness of ˜C k+1 is guaranteed byCorollary 5.2.18. Also, let n k+1 = l k m k+1 n k +1 and C k+1 = Φ −1 (˜C k+1 ).Fix b A ∈ Ω ∗ (C). Using the big-O notation, we havefr(b A ,W k+1 ,g,F)−fr(b A ,W k+1 | [0,nk+1 −n k −1) d,g,F) = O(1/l k), (5.3.6)because these two frequencies differ by the frequency of b A in [n k+1 −n k −1,n k+1 ) d and(n k +1) d ( )n k +1 d= = O(1/ln d k ).k+1 l k m k+1 n k +1The same happens to fr(w k ,w k+1 ,g,F) and fr(w k ,Φ(w k+1 ),g,F), because ∆(g) = 0 for allg ∈ [0,n k+1 −n k −1) d . To simplify citation in the future, we write it down:fr(w k ,w k+1 ,g,F)−fr(w k ,Φ(w k+1 ),g,F) = O(1/l k ). (5.3.7)These estimates imply we can assume, taking l k large enough, that( ) 1−dk 1+d kfr(w k ,w k+1 ,g,F k ) ∈ , , ∀(w k ,w k+1 ,g) ∈ C k ×C k+1 ×Z d .m kd ·|C k | m kd ·|C k |We make a calculation to be used in the next proposition. Fix b A ∈ Ω ∗ (C) and F ∈ F.The main (and simple) observation is: if b A occurs in W k ∈ C k centered at g and w k occursin Φ(w k+1 ) ∈ ˜C k+1 centered at h ∈ [0,l k m k+1 −1) d , then b A occurs in W k+1 ∈ C k+1 centeredat g + n k h. This implies that, if ¯F is a complete residue set modulo F, the cardinality ofS(b A ,W k+1 | [0,nk+1 −n k −1) d,g,F) is equal to∑ ∑|S(b A ,W k ,g −n k h,F)|+Th∈ ¯Fw k ∈C kw k occurring inw k+1 | [0,lk m k+1 −1) dat a vertex ≡ F h= ∑ h∈ ¯Fw k ∈C k∣ ∣∣S(wk,w k+1 | [0,lk m k+1 −1) d,h,F) ∣ ∣∣·|S(bA ,W k ,g −n k h,F)|+T,where T denotes the number of occurrences of b A in W k+1 | [0,nk+1 −n k −1) d not entirely containedin a concatenated element of C k . Observe that 40 ≤ T ≤ d·l k m k+1 ·n d−1 ·n k+1 < dn d−1 · nk+1 2n k,4 For each line parallel to a coordinate axis e iZ between two elements of C k in W k+1 or containing a line ofones, there is a rectangle of dimensions n×···×n×n k+1 ×n×···×n in which b A is not entirely contained ina concatenated element of C k .

Chapter 5. Z d -actions with prescribed topological and ergodic properties 73where n is the length of b A . Dividing(5.3.6), (5.3.7), we getfr(b A ,W k+1 ,g,F) =∣∣S(b A ,W k+1 | [0,nk+1 −n k −1) d,g,F) ∣ ∣∣ by nk+1 d and using( )nk+1 −1 d ∑fr(w k ,w k+1 ,h,F)fr(b A ,W k ,g −n k h,F)n k+1h∈ ¯Fw k ∈C k+O(1/l k−1 ). (5.3.8)We wish to show that fr(b A ,x,F) does not depend on x ∈ X. For this, defineα k (b A ,F) = min{fr(b A ,W k ,g,F); W k ∈ C k ,g ∈ Z d}{β k (b A ,F) = max fr(b A ,W k ,g,F); W k ∈ C k ,g ∈ Z d} .The required property is a direct consequence 5 of the next result.Proposition 5.3.1. If b A ∈ Ω ∗ (C) and F ∈ F, thenlim α k(b A ,F) = lim β k(b A ,F).k→∞ k→∞Proof. By (5.3.3), if l is large such that F l < F, then(F : F l )·α k (b A ,F l ) ≤ α k (b A ,F) ≤ β k (b A ,F) ≤ (F : F l )·β k (b A ,F l ).This means that we can assume F = F l . We estimate α k+1 (b A ,F) and β k+1 (b A ,F) in terms ofα k (b A ,F) and β k (b A ,F), for k ≥ l. As b A and F are fixed, denote the above quantities by α kand β k . Take W k+1 ∈ C k+1 . By (5.3.8),( )nk+1 −1 d ∑fr(b A ,W k+1 ,g,F) ≥ ·α k · fr(w k ,w k+1 ,h,F)+O(1/l k−1 )n k+1h∈ ¯Fw k ∈C k( )nk+1 −1 d= ·α k +O(1/l k−1 )n k+1and, as W k+1 and g are arbitrary, we get( )nk+1 −1 dα k+1 ≥ ·α k +O(1/l k−1 ). (5.3.9)n k+1Equality (5.3.8) also implies the upper bound( )nk+1 −1 d ∑fr(b A ,W k+1 ,g,F) ≤ ·β k ·n k+1==⇒ β k+1 ≤fr(w k ,w k+1 ,h,F)+O(1/l k−1 )h∈ ¯Fw k ∈C k( )nk+1 −1 d·β k +O(1/l k−1 )n k+1( )nk+1 −1 d·β k +O(1/l k−1 ). (5.3.10)n k+15 In fact, just take the limit in the inequality α k (b A,F) ≤ fr(b A,x| Ak ,0,F) ≤ β k (b A,F).

Chapter 5. Z d -actions with prescribed topological and ergodic properties 74Inequalities (5.3.9) and (5.3.10) show that α k+1 and β k+1 do not differ very much from α kand β k . The same happens to their difference. Consider w 1 ,w 2 ∈ C k+1 and g 1 ,g 2 ∈ Z d .Renamingg−n k hbyhin(5.3.8) andconsideringn −k theinverseof n k modulom l , thedifferencefr(b A ,W 1 ,g 1 ,F)−fr(b A ,W 2 ,g 2 ,F) is at most∑From (5.3.5),fr(b A ,W k ,h,F)|fr(w k ,w 1 ,n −k (g 1 −h),F)−fr(w k ,w 2 ,n −k (g 2 −h),F)|h∈ ¯Fw k ∈C k+O(1/l k−1 ).fr(b A ,W 1 ,g 1 ,F)−fr(b A ,W 2 ,g 2 ,F) ≤2d km ld ·|C k |+O(1/l k−1 )∑≤ 2d k +O(1/l k−1 ),fr(b A ,W k ,h,F)h∈ ¯Fw k ∈C kimplying that0 ≤ β k+1 −α k+1 ≤ 2d k +O(1/l k−1 ). (5.3.11)In particular, β k − α k converges to zero as k → +∞. The proposition will be proved if β kconverges. Let us estimate |β k+1 −β k |. On one side, (5.3.10) givesOn the other, by (5.3.9) and (5.3.11),β k+1 −β k ≤ O(1/l k−1 ). (5.3.12)β k+1 −β k ≥ α k+1 −β k( )nk+1 −1 d≥ ·α k −β k +O(1/l k−1 )n k+1( )nk+1 −1 d≥ ·[β k −2d k−1 −O(1/l k−2 )]−β k +O(1/l k−1 )n k+1which, together with (5.3.12), implies that|β k+1 −β k | ≤ 2d k−1 +β k ·[1−= 2d k−1 +O(1/l k−2 ).( ) ]nk+1 −1 d+O(1/l k−2 )n k+1As ∑ d k and ∑ 1/l k both converge, (β k ) k≥1 is a Cauchy sequence, which concludes the proof.

Chapter 5. Z d -actions with prescribed topological and ergodic properties 75From now on, we consider (X,T) as the dynamical system constructed as above. Note thatwe have total freedom to choose C k with few or many elements. This is what controls theentropy of the system.5.3.4 Proof of Theorem 5.1.1By (5.2.2), the topological entropy of the Z d -action (X,T) satisfieslog|C k |h(X,T) ≥ lim ·k→∞ n d kConsider a sequence (ν k ) k≥1 of positive real numbers. In the construction of C k+1 from C k , takel k large enough such that(vii) ν k ·n k+1 ≥ 1.(viii) |˜C k+1 | ≥ |C k | (l km k+1 ) d·(1−ν k ) .These inequalities implylog|C k+1 |≥ log|˜C k+1 |n d k+1 n d k+1≥ (l km k+1 ) d ·(1−ν k )·log|C k |n k+1d≥ (1−ν k ) d+1 · log|C k|n kdand thenlog|C k |n kdk−1∏≥ (1−ν i ) d+1 · log|C k−11| ∏= (1−νnd i ) d+1 ·log2.1i=1i=1If ν ∈ (0,1) is given and (ν k ) k≥1 are chosen also satisfyinglimk→∞i=1k∏(1−ν i ) d+1 = 1−ν,we obtain that h(X,T) ≥ (1−ν)log2 > 0. If, instead of {0,1}, we take C with more elementsand apply the construction verifying (i) to (viii), the topological entropy of the Z d -action is atleast (1−ν)log|C|. We have thus proved Theorem 5.1.1.

Chapter 5. Z d -actions with prescribed topological and ergodic properties 765.4 Proof of Theorems 5.1.2 and 5.1.3Given a finite alphabet C, consider a configuration b A1 : A 1 → C and any A 2 ⊂ A 1 such that|A 2 | ≤ ε|A 1 |. If b A2 : A 2 → C, the element w ∈ Ω A1 (C) defined byw(g) = b A1 (g), if g ∈ A 1 \A 2 ,= b A2 (g), if g ∈ A 2has frequencies not too different from b A1 , depending on how small ε is. In fact, for any c ∈ C,|S(c,b A1 ,g,F)|−|A 2 | ≤ |S(c,w,g,F)| ≤ |S(c,b A1 ,g,F)|+|A 2 |and then |fr(c,b A1 ,g,F)−fr(c,w,g,F)| ≤ ε.Definition 5.4.1. The upper-Banach density of a set P ⊂ Z d is equal tod ∗ |P ∩[r 1 ,r 1 +n 1 )×···×[r d ,r d +n d )|(P) = limsup·n 1 ,...,n d →∞ n 1···n dConsider a set P ⊂ Z d of zero upper-Banach density. We will make l k grow quickly suchthat any pattern of P∩(A k +g) appears as a subconfiguration in an element of C k . Let’s explainthis better. Consider the d-dimensional cubes (A k ) k≥1 that define (X,T). For each k ≥ 1, letà k ⊂ A k be the region containing concatenated elements of C k−1 . Inductively, they are definedas Ã1 = {0} and⋃)à k+1 =(Ãk +n k g +∆(n k g) , ∀k ≥ 1,g∈[0,l k m k+1 ) dwhere ∆ is the function defined in (5.3.2).Lemma 5.4.2. If P ⊂ Z d has zero upper-Banach density, there exists a totally strictly ergodicZ d -action (X,T) with the following property: for any k ≥ 1, g ∈ Z d and b : P∩(A k +g) → {0,1},there exists w k ∈ C k such thatw k (h−g) = b(h), ∀h ∈ P ∩(A k +g).Proof. We proceed by induction on k. The case k = 1 is obvious, since C 1∼ = {0,1}. Supposethe result is true for some k ≥ 1 and consider b : P ∩(A k+1 +g 0 ) → {0,1}. By definition, any0,1 configuration on A k+1 \Ãk+1 is admissible, so that we only have to worry about positionsbelonging to Ãk+1. For each g ∈ [0,l k m k+1 ) d , let)b g : P ∩(Ãk +n k g +∆(n k g)+g 0 → {0,1}

Chapter 5. Z d -actions with prescribed topological and ergodic properties 77)be the restriction of b to P∩(Ãk +n k g +∆(n k g)+g 0 . If ε > 0 is given and l k is large enough,=⇒∣∣P ∩(Ãk+1 +g 0)∣ ∣∣∣∣Ãk+1 +g 0∣ ∣∣ 0 is sufficiently small, ˜z ∈ C k+1 . By its own definition, ˜z satisfies the required conditions.The above lemma is the main property of our construction. It proves the following strongerstatement.Corollary 5.4.3. Let (X,T) be the Z d -action obtained by the previous lemma. For any b : P →{0,1}, there is x ∈ X such that x| P = b. Also, given x ∈ X, A ∈ R and b : P → {0,1}, thereare ˜x ∈ X and n ∈ N such that ˜x| A = x| A and ˜x(g) = b(g) for all g ∈ P\(−n,n) d .Proof. The first statement is a direct consequence of Lemma 5.4.2 and a diagonal argument.For the second, remember that x is the concatenation of elements of C k and lines of zeroes andones, for every k ≥ 1. Consider k ≥ 1 sufficiently large and z k ∈ C k such that x| A occurs inz k . For any z ∈ C k+1 , there is g ∈ Z d such that z| Ak +g = z k . Constructing ˜z from z makingall substitutions described in Lemma 5.4.2, except in the pattern z| Ak +g, we still have that˜z ∈ C k+1 .5.4.1 Proof of Theorem 5.1.2Consider f : X → R given by f(x) = x(0). Then1 ∑ ( )|P ∩(−n,n) d f (T g x) = fr 1,x||P∩(−n,n) d .g∈P∩(−n,n) d

Chapter 5. Z d -actions with prescribed topological and ergodic properties 78For each n ≥ 1, consider the setsΛ n = ⋃ { ( ) }x ∈ X; fr 1,x| P∩(−k,k) d < 1/nk≥nΛ n = ⋃ { ( ) }x ∈ X; fr 1,x| P∩(−k,k) d > 1−1/n .k≥n{}Fixed k and n, the sets x ∈ X; fr(1,x| P∩(−k,k) d) < 1/n and is clearly open, so that the samehappens to Λ n . It is also dense in X, as we will now prove. Fix x ∈ X and ε > 0. Letk 0 ∈ N be large enough so that d(x,y) < ε whenever x| (−k0 ,k 0 ) d = y| (−k 0 ,k 0 ) d. Take y ∈ Xsuch that y| (−k0 ,k 0 ) d = x| (−k 0 ,k 0 ) d and y(g) = 0 for all g ∈ P\(−n,n)d as in Corollary 5.4.3. Asfr(1,y| (−k,k) d) approaches to zero as k approaches to infinity, y ∈ Λ n , proving that Λ n is densein X. The same argument show that Λ n is a dense open set. ThenX 0 = ⋂ n≥1(Λ n ∩Λ n )is a countable intersection of dense open sets, thus residual. For each x ∈ X 0 ,liminfn→∞limsupn→∞1|P ∩(−n,n) d |1|P ∩(−n,n) d |which concludes the proof of Theorem 5.1.2.∑f (T g x) = 0g∈P∩(−n,n) d∑f (T g x) = 1,g∈P∩(−n,n) d5.4.2 Proof of Theorem 5.1.3Choose an infinite set G = {g i } i≥1 in Z d disjoint from P such that P ′ = G∪P ∪{0} also haszero upper-Banach density and let (X,T) be the Z d -action given by Lemma 5.4.2 with respectto P ′ , that is: for every b : P ′ → {0,1}, there exists x b ∈ X such that x b | P ′ = b. Consider{}X 0 = x b ∈ X; b(0) = 0 and b(g) = 1, ∀g ∈ P .This is an uncountable set (it has the same cardinality of 2 G = 2 N ) and, for every x b ∈ X 0 andg ∈ P, the elements T g x b and x b differ at 0 ∈ Z d , implying that x b ∉ {T g x b ; g ∈ P}. Thisconcludes the proof.

Bibliography[1] J. Aaronson, An overview of infinite ergodic theory, Descriptive set theory anddynamicalsystems, LMS Lecture Notes 277 (2000), 1–29.[2] T. Bedford and A. Fisher, Analogues of the Lebesgue density theorem for fractal setsof reals and integers, Proceedings of the London Mathematical Society (3) 64 (1992), no.1, 95–124.[3] V. Bergelson, Ergodic Ramsey Theory - an Update, ErgodicTheoryofZ d -actions, LondonMath. Soc. Lecture Note Series 228 (1996), 1–61.[4] V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden’s and Szemerédi’stheorems, Journal of AMS 9 (1996), no. 3, 725–753.[5] A.S. Besicovitch, On existence of subsets of finite measure of sets of infinite measure,Indagationes Mathematicae 14 (1952), 339–344.[6] J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Publ.Math. IHES69(1989),5–45.[7] Z. Buczolich and D. Mauldin, Divergent square averagesx, Annals of Mathematics 171(2010), no. 3, 1479–1530.[8] K.L. Chung and P. Erdös, Probability limit theorems assuming only the first momentI, Mem. Amer. Math. Soc. 6 (1951).[9] T. Cusick and M. Flahive, The Markov and Lagrange spectra, Mathematical Surveysand Monographs 30, AMS.[10] R.O. Davies, Subsets of finite measure in analytic sets, Indagationes Mathematicae 14(1952), 488–489.[11] K. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, Cambridge(1986).79

Chapter 5. Z d -actions with prescribed topological and ergodic properties 80[12] A. Fisher, Integer Cantor sets and an order-two ergodic theorem, Ergodic Theory andDynamical Systems 13 (1993), no. 1, 45–64.[13] H. Furstenberg, Intersections of Cantor sets and transversality of semigroups, Problemsin analysis (Sympos. Salomon Bochner, Princeton University, 1969), 41–59.[14] H. Furstenberg, Poincaré recurrence and number theory, Bull. Amer. Math. Soc. 5(1981), no. 3, 211–234.[15] H. Furstenberg, Strict ergodicity and transformations of the torus, Amer. J. of Math.83 (1961), 573–601.[16] B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions,Annals of Mathematics 167 (2008), 481–547.[17] F. Hahn and Y. Katznelson, On the entropy of uniquely ergodic transformations, Trans.Amer. Math. Soc. 126 (1967), 335–360.[18] M. Hall, On the sum and product of continued fractions, Annals of Mathematics 48(1947), 966–993.[19] M. Hochman and P. Shmerkin, Local entropy averages and projections of fractalmeasures, to appear in Annals of Mathematics.[20] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst.Hautes Études Sci. Publ. Math. 51 (1980), 137–173.[21] R. Kaufman, On Hausdorff dimension of projections, Mathematika 15 (1968), 153–155.[22] B. Kitchens, Symbolic Dynamics: One-Sided, Two-Sided and Countable State MarkovShifts, Springer (1997).[23] Y. Lima, Z d -actions with prescribed topological and ergodic properties, to appear inErgodic Theory & Dynamical Systems.[24] Y. Lima and C.G. Moreira, A Marstrand theorem for subsets of integers, available athttp://arxiv.org/abs/1011.0672.[25] Y. Lima and C.G. Moreira, A combinatorial proof of Marstrand’s theorem for productsof regular Cantor sets, to appear in Expositiones Mathematicae.[26] Y. Lima and C.G. Moreira, Yet another proof of Marstrand’s theorem, to appear inBulletin of the Brazilian Mathematical Society.

Chapter 5. Z d -actions with prescribed topological and ergodic properties 81[27] J.M. Marstrand, Some fundamental geometrical properties of plane sets of fractionaldimensions, Proceedings of the London Mathematical Society 3 (1954), vol. 4, 257–302.[28] J.M. Marstrand, The dimension of cartesian product sets, Proceedings of the CambridgePhilosophical Society 50 (1954), 198–202.[29] P. Mattila, Hausdorff dimension, projections, and the Fourier transform, Publ. Mat. 48(2004), no. 1, 3–48.[30] P. Mattila and P. Saaranen, Ahlfors-David regular sets and bilipschitz maps, Ann.Acad. Sci. Fenn. Math. 34 (2009), no. 2, 487–502.[31] C.G. Moreira, A dimension formula for images of cartesian products of regular Cantorsets by differentiable real maps, in preparation.[32] C.G. Moreira, Geometric properties of Markov and Lagrange spectra, available athttp://w3.impa.br/∼gugu.[33] C.G. Moreira and J.C. Yoccoz, Stable Intersections of Cantor Sets with Large HausdorffDimension, Annals of Mathematics 154 (2001), 45–96.[34] C.G. Moreira and J.C. Yoccoz, Tangences homoclines stables pour des ensembleshyperboliques de grande dimension fractale, Annales scientifiques de l’ENS 43, fascicule 1(2010).[35] J. Naudts, Dimension of discrete fractal spaces, Journal of Physics A: Math. Gen. 21(1988), no. 2, 447–452.[36] J. Palis, Homoclinic orbits, hyperbolic dynamics and dimension of Cantor sets, ContemporaryMathematics 58 (1987), 203–216.[37] J. Palis and F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinicbifurcations, Cambridge studies in advanced mathematics, Cambridge (1993).[38] J. Palis and F. Takens, Hyperbolicity and the creation of homoclinic orbits, Annals ofMathematics 125 (1987), no. 2, 337–374.[39] J. Palis and J.C. Yoccoz, On the Arithmetic Sum of Regular Cantor Sets, Annales del’Inst. Henri Poincaré, Analyse Non Lineaire 14 (1997), 439–456.[40] R. Pavlov, Some counterexamples in topological dynamics, Ergodic Theory & DynamicalSystems 28 (2008), 1291–1322.

Chapter 5. Z d -actions with prescribed topological and ergodic properties 82[41] Y. Peres and P. Shmerkin, Resonance between Cantor sets, Ergodic Theory & DynamicalSystems 29 (2009), 201–221.[42] K. Petersen, Ergodic Theory, Cambridge University Press (1983).[43] M. Rams, Exceptional parameters for iterated function systems with overlaps, Period.Math. Hungar. 37 (1998), no. 1-3, 111–119.[44] M. Rams, Packing dimension estimation for exceptional parameters, Israel J. Math. 130(2002), 125–144.[45] E. Szemerédi, On sets of integers containing no k elements in arithmetic progression,Acta Arith. 27 (1975), 199–245.[46] B. Weiss, Strictly ergodic models for dynamical systems, Bull. Amer. Math. Soc. 13(1985), no. 2, 143–146.[47] N. Wiener, The ergodic theorem, Duke Math. J. 5 (1939), no. 1, 1–18.