Every finite set of integers is an asymptotic approximate group

TL;DR

The paper proves that any finite set of integers becomes an approximate group with specific parameters when summed sufficiently many times, revealing a universal property of finite integer sets in additive combinatorics.

Contribution

It establishes that every finite set of integers is an asymptotic approximate group with parameters (r, r+1) for all r ≥ 2, a new universal result in additive number theory.

Findings

Every finite set of integers is an asymptotic (r, r+1)-approximate group for all r ≥ 2.

The result applies to all sufficiently large sumsets of finite integer sets.

This generalizes the understanding of approximate groups in additive combinatorics.

Abstract

A set $A$ is an $(r,\ell)$-approximate group in the additive abelian group $G$ if $A$ is a nonempty subset of $G$ and there exists a subset $X$ of $G$ such that $|X| \leq \ell$ and $rA \subseteq X+A$. The set $A$ is an asymptotic $(r,\ell)$-approximate group if the sumset $hA$ is an $(r,\ell)$-approximate group for all sufficiently large integers $h$. It is proved that every finite set of integers is an asymptotic $(r,r+1)$-approximate group for every integer $r \geq 2$.