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

Swaroop Hegde

TL;DR

This paper investigates inverse and stability phenomena for Ruzsa's inequality on triple sumsets in abelian groups. It develops a quantitatively sharp inverse theorem: if |A+A+A| is close to the universal upper bound given |A+A|, then A contains a large almost-3-dissociated subset Y whose triple sum captures most of A+A+A, aligning with Ruzsa's spline construction. The results extend to higher sumsets |(h+1)A| ≤ |hA|^{(h+1)/h} and yield a 99%-stability version of Ruzsa's inequality, plus explicit extremal constructions demonstrating sharpness and limitations of structural descriptions. The paper also provides a generalized inverse theorem for arbitrary h≥2 and develops a robust toolkit combining Loomis-Whitney, Kruskal-Katona, and Clements-Lindström/Macaulay perspectives, along with Plünnecke-type large-subset inequalities. These contributions deepen the understanding of near-extremal sumset behavior and offer quantitative, broadly applicable structure results in additive combinatorics.

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 $1.$

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

TL;DR

This paper investigates inverse and stability phenomena for Ruzsa's inequality on triple sumsets in abelian groups. It develops a quantitatively sharp inverse theorem: if |A+A+A| is close to the universal upper bound given |A+A|, then A contains a large almost-3-dissociated subset Y whose triple sum captures most of A+A+A, aligning with Ruzsa's spline construction. The results extend to higher sumsets |(h+1)A| ≤ |hA|^{(h+1)/h} and yield a 99%-stability version of Ruzsa's inequality, plus explicit extremal constructions demonstrating sharpness and limitations of structural descriptions. The paper also provides a generalized inverse theorem for arbitrary h≥2 and develops a robust toolkit combining Loomis-Whitney, Kruskal-Katona, and Clements-Lindström/Macaulay perspectives, along with Plünnecke-type large-subset inequalities. These contributions deepen the understanding of near-extremal sumset behavior and offer quantitative, broadly applicable structure results in additive combinatorics.

Abstract

Ruzsa's inequality states that for any finite set 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 for some parameter then the set 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 for any We also provide a "99%-stability" version of Ruzsa's inequality, which describes near optimal structures when is very close to
Paper Structure (6 sections, 13 theorems, 48 equations)

This paper contains 6 sections, 13 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Perspectives
  3. Stability using Kruskal-Katona
  4. An inverse result
  5. More Extremal Examples
  6. Higher sumsets

Key Result

Theorem 1.3

Let $A$ be a finite non-empty subset of a commutative group with $|A+A| \leq K|A|.$ Suppose $|A+A+A| \geq \frac{1}{M}K^{3/2}|A|^{3/2},$ where $1\leq M \leq \frac{1}{2}\sqrt{\frac{|A|}{K}}.$ Then there exists $Y \subset A$ such that

Theorems & Definitions (23)

  • Example 1.1: Additively dissociated sets
  • Example 1.2: Ruzsa's construction
  • Theorem 1.3: Inverse result
  • Theorem 1.4: Stability for Theorem \ref{['thm:macaulay_sumsets']}
  • Lemma 2.1: Key observation
  • proof
  • Proposition 2.2: Clements-Lindström clements-lindstrom, Macaulay macaulay
  • Theorem 2.3: Eliahou-Mazumdar eliahou_mazumdar
  • proof
  • Theorem 3.1: Keevash keevash_kk
  • ...and 13 more