Improved stability for the size and structure of sumsets

TL;DR

This paper improves bounds on the size and structure stability of sumsets in integer lattices, providing more precise thresholds for when predictable behavior emerges, advancing understanding in additive combinatorics.

Contribution

It significantly refines the effective bounds for the thresholds of size and structural stability of sumsets, approaching the known lower bounds.

Findings

Enhanced bounds for sumset size thresholds

Refined structural stability thresholds

Closer approximation to theoretical lower bounds

Abstract

Let $A \subset \mathbb{Z}^d$ be a finite set. It is known that the sumset $NA$ has predictable size ($\vert NA\vert = P_A(N)$ for some $P_A(X) \in \mathbb{Q}[X]$) and structure (all of the lattice points in some finite cone other than all of the lattice points in a finite collection of exceptional subcones), once $N$ is larger than some threshold. In previous work, joint with Shakan, the first and third named authors established the first effective bounds for both of these thresholds for an arbitrary set $A$. In this article we substantially improve each of these bounds, coming much closer to the corresponding lower bounds known.