- Published 2008

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 ∈ L2(G). We prove an analogous result for functions f ∈ A(G), where A(G) := {f ∈ L1(G) : ‖f̂‖1 < ∞} equipped with the norm ‖f‖A(G) := ‖f̂‖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 ‖χA‖A(Z/pZ) ≫ε (log p) 1/3−ε, and we improve this to ‖χA‖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 ‖χ̂A‖A(Z/pZ) ≪ log p.

@inproceedings{Sanders2008arXI,
title={ar X iv : m at h / 06 05 52 2 v 1 [ m at h . C A ] 1 8 M ay 2 00 6 THE LITTLEWOOD - GOWERS PROBLEM},
author={T. Sanders},
year={2008}
}