The Littlewood-Gowers problem

Abstract

Abstract. The paper has two main parts. To begin with suppose that G is a compact Abelian group. Chang's Theorem can be viewed as a structural refinement of Bessel's inequality for functions f ∈ L 2 (G). We prove an analogous result for functions f ∈ A(G), where A(G) is the space {f ∈ L 1 (G) : f 1 < ∞} endowed with the norm f A(G) := f 1 , and generalize this to the approximate Fourier transform on Bohr sets.As an application of the first part of the paper we improve a recent result of Green and Konyagin: Suppose that p is a prime number and A ⊂ Z/pZ has density bounded away from 0 and 1 by an absolute constant. Green and Konyagin have shown that χ A A(Z/pZ) ≫ε (log p) 1/3−ε ; we improve this to χ A A(Z/pZ) ≫ε (log p) 1/2−ε . To put this in context it is easy to see that if A is an arithmetic progression then χ A A(Z/pZ) ≪ log p.

A quantitative version of the Beurling-Helson theorem

“…After the discretization of the problem, an appropriate upper bound for dimension allows us to consider the values of the function ϕ only on a small subset of the lattice rather than at all lattice points. We also slightly improve the estimate by using results of Sanders [12] on lower bounds for the Wiener norms of the characteristic functions of large subsets of Z p (see Section 3) in the case of prime p. As a byproduct, we obtain the best possible lower estimates for the Wiener norms of the characteristic functions of small subsets of Z p .…”

“…In the theory of Bohr sets, the following result plays an important role (see, e.g., Lemma 6.2 in [12]). G (B(Γ, δ)) δ d holds.…”

Dependent random choice

“…This question was answered affirmatively by Green and Konyagin [54]. The best current bound is due to Sanders [76], who showed that there is x ∈ Z p such that ||A∩(A+x)|−p/4| = O(p/ log 1/3 p). Both these results used Fourier analysis.…”

Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem

“…This improvement would allow us to replace the right-hand side in the condition of our theorem with o((log log |n|) 1/4 ). Apparently, it is easier to obtain an improvement of estimate (2) with the replacement of the exponent 1/3 with 1/2; in this connection, see the work of Sanders [15]. This would allow us to replace the exponent 1/12 in our theorem with 1/8.…”