A generalization of the asymptotic formula in Waring's problem with ...

THE VINOGRADOV l 2 MAXIMAL THEOREM FOR HIGHER ORDER
SPHERES
K HUGHES
Contents
1. Introduction 1
2. The main l p inequality 2
3. Aprroximating by the Hardy-Littlewood multiplier HL 2
3.1. The major arcs 4
3.2. The minor arcs 6
4. The Hardy-Littlewood multiplier 7
5. Extending the result to p < 2 8
6. Continuous analogue: Convex hypersurfaces: Fourier tranforms and averages 8
References 9
1. Introduction
Throughout fix k ≥ 2 an integer. Define the ”k-sphere” norm |x| k for x ∈ R d to be ∑ d
i=1 |x i| k
and the indicator function of the discrete k-sphere to be σ r = 1 {x∈Z d :|x| k =r k } of radius r which
contains N d,k (r) lattice points. The dimension d will always be greater than k.
A r f = 1
N(r) σ r ∗ |f|
A R f = A r f
sup
R/2≤r0
Remark 1.1. The supremum above is taken over all positive r such that N(r) ≠ 0 e.g. only k-spheres
with lattice points on them. For large enough dimension, this is equivalent to r k ∈ N.
Using Vinogradov type estimates and the methods of Magyar([6]) and Magyar–Stein–Wainger
aka MSW ([8]), I show
Theorem 1. For k ≥ 3, A ∗ is a bound operator on l 2 (Z d ) provided d > 11k 3 log k.
Interpolating with a trivial l 1 bound for the dyadic maximal operator (see [6]), I can improve
the range of l p spaces to
Theorem 2. For k ≥ 3, A ∗ is a bound operator on l p (Z d ) for p > 1 +
d > 11k 3 log k.
In [8], MSW studied this problem when k = 2 proving the sharp result:
11k3 log k
2d−11k 3 log k
Magyar–Stein–Wainger. For k = 2 and d ≥ 5, ‖A ∗ f‖ l p ‖f‖ l p for all p > d
d−2 .
1
provided

Subsequently, in [7], Magyar extended the MSW result to averages over nondegenerate positive
integral forms proving the same result with d > k2 k−1 . This includes the family of hypersurfaces
considered here. This note improves the dimension from exponential to polynomial by using Vinogradov
type bounds for exponential sums. The proof relies on a Hardy-Littlewood decomposition
as in [6, 8] to approximate the multiplier of the convolution operator in the l 2 norm and showing
that both the approximant and approximation are l 2 bounded. We note that in the context of the
circle method, Magyar–Stein–Wainger corresponds to the sums of squares in Waring’s problem,
while this note corresponds to the sums of k th powers in Waring’s problem.
Remark 1.2. Testing the maximal operator on the delta function, we expect that the maximal
operator is bounded on l p for p >
d
d−k
as in MSW. However, the range I obtain is far away from
this and to improve it using the method in this paper, I would need to improve Vinogradov estimates
for exponential sums which is a difficult, open problem in number theory. You might think that
one could improve the l 1 inequality, but if I could, then I could probably do this in 2 dimensions
and this would give me a better range of l p spaces than is possible.
2. The main l p inequality
We will make use of the following inequality below. If T r is a convolution operator with multiplier
m r (ξ) = ∫ I α R(t, ξ)e(−r k t) dt for some measurable subset of the torus. Then
∫
sup |T r f|(x) =
∣ α R (t, ξ)e(−r
R/2≤r

where N k (r) = #{y : |y| k = r k }. For sufficiently large d, we know that N(r) ∼ d,k r d−k . So we may
replace the averages A r with
∑
r d−k f(x − y)
|y| k =r k
also denoted by A r . Now A r is a convolution operator with multiplier r k−d a r (ξ) where
a r (ξ) =
∑
e(x · ξ)
|x| k =r k
And since ∫ T
e(t(x − y))dt is 1 or 0 depending on whether x = y or not, we can rewrite this as
a r (ξ) = ∑ ∫
e((|x| k − r k )t + x · ξ) dt
x∈Z d
and furthermore we can dampen the sum by inserting factors of e −2πɛ|x|k to get
a r (ξ) = e ∑ ∫
2πɛrk e(|x| k (t + iɛ) − tr k + xξ) dt
x∈Z d
T
T
Let H z (ξ) = ∑ x∈Z d e(|x|k z + xξ), then exchanging the sum and integral above (which is possible
by the dampening) a r (ξ) = e
∫T 2πɛrk e(−trk )H t+iɛ (ξ) dt. For a level R k−1 > 0, make a Farey
dissection on T (which I identify with [0, 1] via the exponential map e(·)) of level R k−1 . This
yields a sequence of intervals Ī(a/q) = {θ ∈ [0, 1] : −β/qRk−1 ≤ θ − a/q ≤ α/qR k−1 } where
α(a/q, R k−1 ), β(a/q, R k−1 ) ≈ 1. I(a/q) = Ī(a/q) − a/q. Decompose a r with respect to this dissection:
a r (ξ) =
∑ ∑
a a/q
r (ξ)
q≤R k−1 (a,q)=1
with
∫
a a/q
r (ξ) = e 2πɛrk e(r k a/q) e(−tr k )H z (t − a/q + iɛ, ξ) dt
I(a/q)
For a parameter Y which we optimize below, split our multiplier into pieces from major arcs
and pieces from minor arcs
a Major
r
a minor
r (ξ) =
(ξ) = ∑ q

Basic Weyl sum bound. For all a ∈ Z/qZ ∗ , x ∈ Z,
1
q∑
e(b k a/q + bx/q) q −1/k .
q
n=1
Remark 3.1. One can actually improve this using an inequality of Hua to ≪ δ q δ− 1 2 for all δ > 0.
If we let,
∫
J r (ξ − x/q) = ˜h z (x/q − ξ)e(−r k t) dt,
then we can write
I(a/q)
a a/q
r (ξ) = e 2πɛrk e(r k a/q) ∑ x
G(a/q, x)J r (ξ − x/q).
Remark 3.2. We will always choose ɛ = R −k so that the factor e 2πɛrk ≪ 1.
3.1. The major arcs. Introduce the approximating multipliers b a/q
r , c a/q
r defined as follows:
b a/q
r (ξ) = e 2πɛrk e(r k a/q) ∑ x
G(a/q, x)Ψ(qξ − x)J r (ξ − x/q)
c a/q
r (ξ) = e 2πɛrk e(r k a/q) ∑ x
G(a/q, x)Ψ(qξ − x)I r (ξ − x/q)
where I r = ∫ ˜h R z (x/q − ξ) is obtained from J r by extending the range of integration from I(a/q)
to R. Using these we define multipliers b r , c r with convolution operators Br
Major , Cr
Major by
b Major
r (ξ) = ∑ ∑
b a/q
r (ξ)
q 0
such that for all f ∈ l 2 and R > 0,
∥
sup |A Major
r f − Cr Major
f| ∥ R d−k−θ ‖f‖ l 2
R/2≤r

(2) ˜h z (ξ − x/q) ≪ |z| − d
2k−2 |ξ − l/q|
−d k−2
2k−2 e
−K·|qξ−l| k−1
for some constant K = K d,k depending
only on dimension and degree.
∥
Lemma 3.2. ∥sup R/2≤r 1/2. Then we have
∑
G(a, q, x)(1 − Ψ(qξ − x))˜h z (ξ − x/q) =
x
∑
|qξ−x|>1/2
≪ sup
x
≪ sup
x
≪ sup
x
≪ sup
x
k
G(a, q, x)(1 − Ψ(qξ − x))˜h z (ξ − x/q)
|G(a, q, x)|
|G(a, q, x)|
∑
|qξ−x|>1/2
∑
|qξ−x|>1/2
|G(a, q, x)| · |z| −
|G(a, q, x)| · |z| −
|˜h z (ξ − x/q)|
|z| − d
2k−2 |ξ − l/q|
−d k−2
d d(k−2)
2k−2 q 2k−2
d d(k−2)
2k−2 q 2k−2
≪ sup |G(a, q, x)| · (qR k−1 ) d d(k−2)
2k−2 q 2k−2
x
= sup |G(a, q, x)| · R d/2 q d/2
x
≪ R d/2 q d/2−d/k
∑
|qξ−x|>1/2
k
2k−2 e
−K·|qξ−l| k−1
k−2
−d
|qξ − l|
Summing over a ∈ Z/qZ ∗ and 1 ≤ q ≤ R, we get a bound of ≪ R d+2−d/k ). We need power
saving in R, so we choose d > k(k + 2).
∥
Lemma 3.3. ∥sup R/2≤r

We sum over a and q to find
R∑ ∑
∥
Choose d > k(k + 2).
q=1 (a,q)=1
sup
R/2≤r

Choosing d such that
d
11k 2 log k > k or more simply, d > 11k3 log k, we get
Corollary 3.1. For d > 11k 3 log k, there exists a θ(k, d) > 0 such that for all f ∈ l 2 and R > 0,
∥
sup |A minor
r f| ∥ ≪ R d−k−θ ‖f‖ l 2
R/2≤r0
Lemma 4.1. ‖C ∗ f‖ l 2 ‖f‖ l 2 where the implied constant depends only on the degree k and the
dimension d.
Proof. Recall that
Ĉ r f(ξ) =
∞∑
By the triangle inequality, we have
To prove this, we will show that
Then
∑
∑
q=1 a∈Z/qZ ∗
‖C ∗ f‖ l 2 ≤
∥
If d > 2k, then the sum converges and
∥C∗
a/q
C a/q
r
to
x∈Z d G(a, q, x)Ψ(qξ − x)˜dσ r (ξ − x/q)
∞∑ ∑ ∥ ∥∥C a/q ∥
∗ f
a∈Z/qZ ∗
q=1
f∥ q−d/k ‖f‖
l 2 l 2 .
∥
l 2
∑
∞ ∑
‖C ∗ f‖ l 2 k,d
q −d/k ‖f‖ l 2
q=1 a∈Z/qZ ∗
∞∑
δ,d,k ‖f‖ l 2 q −d/k+1 .
q=1
‖C ∗ f‖ l 2 k,d ‖f‖ l 2 .
To prove the aforementioned inequality, we first separate our study of Cr
a/q f into a finite part
and a continuous part by writing
̂
C a/q (ξ) = ∑ G(a, q, x)Ψ(qξ − x)Ψ 1 (qξ − x)dσ r (ξ − x/q)
x∈Z d
r
⎛
⎞ ⎛
= ⎝ ∑ G(a, q, x)Ψ(qξ − x) ⎠ ⎝ ∑ Ψ 1 (qξ − x)d˜σ r (ξ − x/q) ⎠
x∈Z d x∈Z d
⎞

and n a/q
r with
convolution operators Sr
a/q and Tr
a/q respectively, so that Cr
a/q f = Sr
a/q Tr
a/q f = Tr
a/q Sr
a/q f (note
that S implicitly depends on r, but when we bound it later, we can bound it independently of r).
This means that we only need to show that each maximal operator is bounded on l 2 for arbitrary
functions; the norm would be bounded by the product of these norms. T∗
a/q is bounded using a
vector-valued version of a theorem of Bruna–Nagel–Wainger (see below) while S∗
a/q is bounded by
For each r, our multiplier is the product of two commuting multipliers, say m a/q
r
using Plancherel’s theorem (include Stein-Wainger folk lemma later). Given the support of Ψ and
known bounds for Weyl sums, we can bound the first part by ≪ q −d/k .
□
Remark 4.1. Combining the Basic Weyl sum bound with the Stein–Wainger folk lemma we can
actually say more (at least about the arithmetic part): the Hardy–Littlewood multiplier is bounded
on l p for p >
d
d−k . 5. Extending the result to p < 2
By the previous section we already know that the Hardy–Littlewood multiplier is bounded on
l p for p >
d
d−k
, so we want to understand the range of p for which the error terms are bounded
on l p . The methods above are special l 2 , so we’ll combine these by interpolation with a trivial l 1
bound. The trivial l 1 bound for the dyadic maximal operator is R k . The l 2 bound is R −θ for some
θ depending on k and d. Since k is fixed, we see that θ → ∞ as d → ∞ so that the error terms are
bounded in l p for p > p 0 (d) where p 0 (d) → 1 while d → ∞.
More precisely, interpolation tells us that the bound for the dyadic maximal operator will be
R −ɛ(p) for some positive ɛ(p) when p > 2(θ+k)
2θ+k = 1 + k
2θ+k
. Examining the proof we see that the
θ we can choose for the error terms in studying the major arcs is d k
− 2 − k while the θ we can
choose for the error terms in studying the minor arcs is − k. We must use the smallest
which is θ(k, d) =
p > 1 +
d
d
11k 2 log k
11k 2 log k − k for d > 11k3 log k. Plugging this into our bound above, we get
k
−k+2d/(11k 2 log k) = 1 + 11k3 log k
2d−11k 3 log k for d > 11k3 log k.
6. Continuous analogue: Convex hypersurfaces: Fourier tranforms and averages
Let S be a smooth, compact, convex hypersurface in R d+1 for d > 1 of finite type and σ its
induced Lebesgue surface measure (a will be arbitrary but all implicit constants may depend on
it). Furthermore, let dσ 0 (x) = a(x)dσ(x) where a is a smooth, compactly supported function in a
neighborhood of x 0 ∈ S. For x ∈ S, denote the outward unit normal to S at x by v x and if T x is
the tangent plane at x, then for δ > 0, define the δ-ball about x as
Define the averages
and the associated maximal function
Bruna–Nagel–Wainger. As t → ∞,
B(x, δ) = {y ∈ S : dist(y, T x ) < δ}.
∫
A t f(x) =
S
f(x − ty)dσ 0 (y)
Mf(x) = sup |A t f(x)|.
t>0
̂dσ(tv x ) σ(B(x, t −1 ))
Interpolation. If ̂dσ(ξ) |ξ| −γ for large ξ and some γ > 1/2, then M is a bounded operator on
L p for p > 1 + (2γ) −1

Define the norm |x| k = ∑ d+1
i=1 |x i| k and the d-dimensional hypersurface S d k := {x : |x|k = 1}. S d k
is finite type of order k and for the poles x = (0, . . . , 0, ±1), etc., we have
σ(B(x, δ)) δ d/k .
Considering a smooth partition of unity on the hypersurface and the above theorem, we obtain
Corollary 6.1. For d > 2k, M is a bounded operator on L p for p > 1 + (2d/k) −1 = 2d+k
2d .
Remark 6.1. I need to check this because it does not match with the spherical case
2d−2 =
This would give p > d+1
d
d
, but this exponent is <
d−1
which is the right exponent for the sphere.
Remark 6.2. While these reults may not be sharp in the continuous setting, this is sufficient for
our purposes.
References
[1] M. Avdispahic and L. Smajlovic, On maximal operators on k-spheres in Z n Proceedings of the American Mathematical
Society, Vol. 134, No. 7, Pages 2125-2130, 2006.
[2] Bruna, Nagel and Wainger Something about convex hypersurfaces .
[3] H. Davenport, Analytic methods for Diophantine equations and inequalities Cambridge University Press, 2nd ed.,
2005;
[4] G. Hardy and J. Littlewood, Some problems of ’Partitio Numerorum 1: A new solution of Waring’s problem
Hardy’s Collected works, .
[5] H. Iwaniec and E. Kowalski, Analytic number theory American Mathematical Society, Colloquium Publications,
V. 53, 2004.
[6] A. Magyar, L p bounds for spherical maximal operators on Z n Revista Matematica Iberoamericana, Vol. 13, No.
2, 1997;
[7] A. Magyar, Diophantine equations and ergodic theorems Revista Matematica Iberoamericana, Vol. 13, No. 2,
1997;
[8] A. Magyar, E. Stein and S. Wainger Discrete analogues in harmonic analysis: Spherical averages Annals of
Mathematics, 2005;
[9] H. Montgomery, 10 Lectures on the interface between analytic number theory and harmonic analysis Conference
Board of the Mathematical Sciences, Regional Conference Series in Mathematics, Number 84, 1994.
2d
d
d−1 .