An elementary additive doubling inequality

TL;DR

This paper proves a simple additive combinatorics inequality showing that sets with large doubling in Abelian groups contain large subsets with small doubling, revealing structural properties of such sets.

Contribution

It introduces an elementary inequality linking large doubling sets to subsets with small doubling, advancing understanding in additive combinatorics.

Findings

Sets with large doubling contain large subsets with small doubling

The inequality applies to subsets of Abelian groups

Provides a new elementary tool for additive combinatorics

Abstract

We prove an elementary additive combinatorics inequality, which says that if $A$ is a subset of an Abelian group, which has, in some strong sense, large doubling, then the difference set A-A has a large subset, which has small doubling.