Normed versus topological groups: Dichotomy and duality

102 N. H. Bingham and A. J. OstaszewskiTheorem 9.3 (Conjunction Theorem). For V, W Baire non-meagre (resp. measurablenon-null) in a group G equipped with either the norm or the density topology, there isa ∈ G such that V ∩ (aW ) is Baire non-meagre (resp. measurable non-null) and for anynull sequence z n → e G and quasi all (almost all) t ∈ V ∩ (aW ) there exists an infiniteM t such that{tz m : m ∈ M t } ⊂ V ∩ (aW ).Proof. In either case applying Theorem 9.2, for some a the set T := V ∩ (aW ) is Bairenon-meagre (resp. measurable non-null). We may now apply the Kestelman-Borwein-Ditor Theorem to the set T. Thus for almost all t ∈ T there is an infinite M t suchthat{tz m : m ∈ M t } ⊂ T ⊂ V ∩ (aW ).See [BOst-KCC] for other forms of countable conjunction theorems. The last resultmotivates a further strengthening of generic subuniversality (compare Section 6).Definitions. Let S be generically subuniversal (=null-shift-compact). (See the definitionsafter Th. 7.7.)1. Call T similar to S if for every null sequence z n → e G there is t ∈ S ∩ T and M t suchthat{tz m : m ∈ M t } ⊂ S ∩ T.Thus S is similar to T and both are generically subuniversal.Call T weakly similar to S if if for every null sequence z n → 0 there is s ∈ S and M ssuch that{sz m : m ∈ M s } ⊂ T.Thus again T is subuniversal (=null-shift-precompact).2. Call S subuniversally self-similar, or just self-similar (up to inversion-translation), iffor some a ∈ G and some T ⊂ S, S is similar to aT −1 .Call S weakly self-similar (up to inversion-translation) if for some a ∈ G and some T ⊂ S,S is weakly similar to aT −1 .Theorem 9.4 (Self-similarity Theorem). In a group G equipped with either the norm orthe density topology, for S Baire non-meagre (or measurable non-null), S is self-similar.Proof. Fix a null sequence z n → 0. If S is Baire non-meagre (or measurable non-null),then so is S −1 ; thus we have for some a that T := S ∩(aS −1 ) is likewise Baire non-meagre(or measurable non-null) and so for quasi all (almost all) t ∈ T there is an infinite M tsuch that{tz m : m ∈ M t } ⊂ T ⊂ S ∩ (aS −1 ),as required.