The structure of higher sumsets

TL;DR

This paper characterizes the structure of higher sumsets of finite integer sets with gcd 1, showing they decompose into a union of two sets separated by a long block, with precise estimates and stability analysis.

Contribution

It extends classical results by providing a detailed structural description of large multiple sumsets, including sharp bounds and stability classifications.

Findings

Sumsets decompose into 'head', 'tail', and a long separating block.

Sharp estimates for the length of the separating block.

Classification of sets where bounds cannot be improved.

Abstract

Merging together a result of Nathanson from the early 70s and a recent result of Granville and Walker, we show that for any finite set $A$ of integers with $\min(A)=0$ and $\gcd(A)=1$ there exist two sets, the "head" and the "tail", such that if $m\ge\max(A)-|A|+2$, then the $m$-fold sumset $mA$ consists of the union of these sets and a long block of consecutive integers separating them. We give sharp estimates for the length of the block, and investigate the corresponding stability problem classifying those sets $A$ for which the bound $\max(A)-|A|+2$ cannot be substantially improved.