A tight structure theorem for sumsets

TL;DR

This paper proves a structural theorem for sumsets of finite integer sets, confirming a recent conjecture and classifying extremal cases, advancing understanding in additive combinatorics.

Contribution

It establishes a precise structure theorem for sumsets when the number of summands exceeds a certain bound, and classifies sets where this bound is tight.

Findings

Sumset $NA$ has a specific structure when $N \\geq b - \\ell$

Confirmed a conjecture by Shakan and the first author

Identified sets where the bound cannot be improved

Abstract

Let $A = \{0 = a_0 < a_1 < \cdots < a_{\ell + 1} = b\}$ be a finite set of non-negative integers. We prove that the sumset $NA$ has a certain easily-described structure, provided that $N \geqslant b-\ell$, as recently conjectured by Shakan and the first author. We also classify those sets $A$ for which this bound cannot be improved.