An inverse and a stability result for Ruzsa's inequality on triple sumsets

TL;DR

This paper establishes an inverse and stability version of Ruzsa's inequality for triple sumsets, characterizing the structure of sets that nearly attain the inequality's bounds and extending results to higher sumsets.

Contribution

It proves an inverse theorem for Ruzsa's inequality, showing near extremal sets resemble known constructions, and extends the results to higher sumsets with a stability version.

Findings

Sets with large triple sumsets resemble Ruzsa's extremal examples.

The inverse result is likely optimal in a qualitative sense.

A stability version describes near extremal structures when the inequality is nearly tight.

Abstract

Ruzsa's inequality states that $|A+A+A| \leq |A+A|^{3/2}$ for any finite set $A$ in a commutative group. Ruzsa has constructed examples showing that this inequality is sharp asymptotically, up to a constant factor. We prove an inverse result which says that if $|A+A+A| \geq \frac{1}{M} |A+A|^{3/2}$ for some parameter $M,$ then the set $A$ resembles the sets in Ruzsa's construction. We then construct more families of examples which suggest that our inverse result is likely best possible qualitatively. The method extends to give an inverse result for a higher sumset analogue of Ruzsa's inequality, namely $|(h+1)A| \leq |hA|^{\frac{h+1}{h}}$ for any $h\geq 2.$ We also provide a "99%-stability" version of Ruzsa's inequality, which describes near optimal structures when $M$ is very close to $1.$