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On the number of sets with small sumset

Dingyuan Liu, Letícia Mattos, Tibor Szabó

TL;DR

This work analyzes the number of $s$-subsets $A$ of an $n$-element set $Y$ in an abelian group with a small sumset $|A+A|\le m$. It develops an efficient container framework, combining a graph-based sumrise procedure with a hypergraph-based Sunset refinement, to produce a nearly optimal collection of containers and derive tight counting bounds of the form $2^{o(s)}\binom{\frac{m+\beta}{2}}{s}$ where $\beta=\beta_G(m+o(m))$. The authors extend and sharpen prior results (notably Campos) and nearly resolve a conjecture of Alon–Balogh–Morris–Samotij, while also establishing typical-structure results: a random set from $\mathcal{F}_{[n]}(m,s)$ is with high probability almost contained in an arithmetic progression of size $m/2+o(m)$. The methods yield sharp, parameter-robust bounds in arbitrary abelian groups and illuminate the typical geometry of small sumsets via a streamlined container methodology that handles both symmetric and asymmetric sumset settings.

Abstract

We investigate subsets with small sumset in arbitrary abelian groups. For an abelian group $G$ and an $n$-element subset $Y \subseteq G$ we show that if $m \ll s^2/(\log n)^2$, then the number of subsets $A \subseteq Y$ with $|A| = s$ and $|A + A| \leq m$ is at most \[2^{o(s)}\binom{\frac{m+β}{2}}{s},\] where $β$ is the size of the largest subgroup of $G$ of size at most $\left(1+o(1)\right)m$. This bound is sharp for $\mathbb{Z}$ and many other groups. Our result improves the one of Campos and nearly bridges the remaining gap in a conjecture of Alon, Balogh, Morris, and Samotij. We also explore the behaviour of uniformly chosen random sets $A \subseteq \{1,\ldots,n\}$ with $|A| = s$ and $|A + A| \leq m$. Under the same assumption that $m \ll s^2/(\log n)^2$, we show that with high probability there exists an arithmetic progression $P \subseteq \mathbb{Z}$ of size at most $m/2 + o(m)$ containing all but $o(s)$ elements of $A$. Analogous results are obtained for asymmetric sumsets, improving results by Campos, Coulson, Serra, and Wötzel. The main tool behind our results is a more efficient container-type theorem developed for sets with small sumset, which gives an essentially optimal collection of containers. The proof of this combines an adapted hypergraph container lemma, that caters to the asymmetric setup as well, with a novel ``preprocessing'' graph container lemma, which allows the hypergraph container lemma to be called upon significantly less times than was necessary before.

On the number of sets with small sumset

TL;DR

This work analyzes the number of -subsets of an -element set in an abelian group with a small sumset . It develops an efficient container framework, combining a graph-based sumrise procedure with a hypergraph-based Sunset refinement, to produce a nearly optimal collection of containers and derive tight counting bounds of the form where . The authors extend and sharpen prior results (notably Campos) and nearly resolve a conjecture of Alon–Balogh–Morris–Samotij, while also establishing typical-structure results: a random set from is with high probability almost contained in an arithmetic progression of size . The methods yield sharp, parameter-robust bounds in arbitrary abelian groups and illuminate the typical geometry of small sumsets via a streamlined container methodology that handles both symmetric and asymmetric sumset settings.

Abstract

We investigate subsets with small sumset in arbitrary abelian groups. For an abelian group and an -element subset we show that if , then the number of subsets with and is at most where is the size of the largest subgroup of of size at most . This bound is sharp for and many other groups. Our result improves the one of Campos and nearly bridges the remaining gap in a conjecture of Alon, Balogh, Morris, and Samotij. We also explore the behaviour of uniformly chosen random sets with and . Under the same assumption that , we show that with high probability there exists an arithmetic progression of size at most containing all but elements of . Analogous results are obtained for asymmetric sumsets, improving results by Campos, Coulson, Serra, and Wötzel. The main tool behind our results is a more efficient container-type theorem developed for sets with small sumset, which gives an essentially optimal collection of containers. The proof of this combines an adapted hypergraph container lemma, that caters to the asymmetric setup as well, with a novel ``preprocessing'' graph container lemma, which allows the hypergraph container lemma to be called upon significantly less times than was necessary before.
Paper Structure (6 sections, 19 theorems, 119 equations, 2 algorithms)

This paper contains 6 sections, 19 theorems, 119 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Container theorems and proof idea
  3. sumrise: the first container algorithm
  4. Sunset: the second container algorithm
  5. Counting sets with small sumset
  6. The typical structure of sets with small sumset

Key Result

Theorem 1.2

Let $n, m=m(n), s=s(n) \in \mathbb{N}_{\ge 1}$ be integers such that $2s \le m \ll \frac{s^2}{(\log n)^2}$ as $n \rightarrow \infty$. Let $G$ be an arbitrary abelian group and $Y$ be an $n$-element subset of $G$For a meaningful statement we actually need to speak about sequences of groups $G_n$ and where $\beta = \beta_G\left(m+o(m)\right)$.

Theorems & Definitions (39)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 29 more