Covering the large spectrum and generalized Riesz products

TL;DR

This paper explores how advanced variants of Chang's Lemma, crucial in additive combinatorics, can be derived from entropy maximization techniques, leading to stronger results in the field.

Contribution

It introduces a novel approach to deriving Chang's Lemma variants using entropy maximization, enhancing the quantitative bounds in additive combinatorics.

Findings

Bloom's variant yields the strongest known bounds for Roth's theorem

Entropy maximization provides a new framework for deriving combinatorial theorems

The approach unifies different versions of Chang's Lemma

Abstract

Chang's Lemma is a widely employed result in additive combinatorics. It gives bounds on the dimension of the large spectrum of probability distributions on finite abelian groups. Recently, Bloom (2016) presented a powerful variant of Chang's Lemma that yields the strongest known quantitative version of Roth's theorem on 3-term arithmetic progressions in dense subsets of the integers. In this note, we show how such theorems can be derived from the approximation of probability measures via entropy maximization.