On Isoperimetric Stability

TL;DR

This paper establishes a lower bound on the size of subsets in abelian groups with small edge boundaries, introduces bounds on independent subsets of difference sets, and provides an auxiliary estimate for downsets in integer lattices.

Contribution

It presents a new isoperimetric inequality for abelian groups, sharp bounds for independent difference sets, and an auxiliary result on the structure of downsets in integer lattices.

Findings

A non-empty subset with small edge boundary must be large, with a bound involving the smallest order of elements.

Derived an upper bound for the size of the largest independent subset of popular differences.

Established an average weight bound for downsets in  in terms of their size.

Abstract

We show that a non-empty subset of an abelian group with a small edge boundary must be large; in particular, if $A$ and $S$ are finite, non-empty subsets of an abelian group such that $S$ is independent, and the edge boundary of $A$ with respect to $S$ does not exceed $(1-\gamma)|S||A|$ with a real $\gamma\in(0,1]$, then $|A| \ge 4^{(1-1/d)\gamma |S|}$, where $d$ is the smallest order of an element of $S$. Here the constant $4$ is best possible. As a corollary, we derive an upper bound for the size of the largest independent subset of the set of popular differences of a finite subset of an abelian group. For groups of exponent $2$ and $3$, our bound translates into a sharp estimate for the additive dimension of the popular difference set. We also prove, as an auxiliary result, the following estimate of possible independent interest: if $A \subset \mathbb Z^n$ is a finite, non-empty downset then, denoting by $w(a)$ the number of non-zero components of the vector $a\in A$, we have \[\frac1{|A|} \sum_{a\in A} w(a) \le \frac12\, \log_2 |A|.\]