On common energies and sumsets II

TL;DR

This paper explores the relationship between set sum sizes and common energies, establishing a sharp subexponential bound and improving the arithmetic regularity lemma's parameter dependence.

Contribution

It introduces a sharp subexponential bound linking the doubling constant and minimal common energy, and provides an improved proof of the arithmetic regularity lemma.

Findings

Established a subexponential dependence between doubling constant and common energy.

Proved a version of the arithmetic regularity lemma with better parameter dependence.

Analyzed the relationship between sumset size and subset energies.

Abstract

We continue to study the relationship between the size of the sum of a set and the common energy of its subsets. We find a rather sharp subexponential dependence between the doubling constant of a set $A$ and the minimal common energy taken over all partitions of $A$ into two disjoint subsets. As an application, we give a proof of the well--known arithmetic regularity lemma with better dependence on parameters.