The relative sizes of sumsets and difference sets

TL;DR

This paper characterizes when the inequality relating sumset and difference set sizes becomes equality, revealing that equality occurs only for cosets of finite subgroups, and discusses implications for related inequalities.

Contribution

It precisely identifies the conditions for equality in the sumset-difference set size inequality and explores the potential for improving the exponent 1/2 in the inequality.

Findings

Equality holds iff A is a coset of a finite subgroup.

Both doubling and difference constants equal 1 at equality.

Plünnecke's inequality is strict when the doubling constant exceeds 1.

Abstract

Let A be a finite subset of a commutative additive group Z. The sumset and difference set of A are defined as the sets of pairwise sums and differences of elements of A, respectively. The well-known inequality $\sigma(A)^{1/2} \leq \delta(A) \leq \sigma(A)^2,$ where $\sigma(A)=\frac{|A+A|}{|A|}$ is the doubling constant of A and $\delta(A)=\frac{|A-A|}{|A|}$ is the difference constant of A, relates the relative sizes of the sumset and difference set of A. The exponent 2 in this inequality is known to be optimal, for the exponent 1/2 this is unknown. We determine those sets for which equality holds in the above inequality. We find that equality holds if and only if A is a coset of some finite subgroup of Z or, equivalently, if and only if both the doubling constant and difference constant are equal to 1. This implies that there is space for possible improvement of the exponent 1/2 in the inequality. We then use the derived methods to show that Pl\"unnecke's inequality is strict when the doubling constant is larger than 1.