Improved stability for the size and structure of iterated sumsets in $\mathbb{Z}^d$

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, the first effective bounds for both of these thresholds were established, for an arbitrary set $A$. In this article we substantially improve each of these bounds, coming much closer to the corresponding lower bounds known.Mathematics Subject Classifications: 11P21, 05B10, 11B13, 11P70, 05A16Keywords: Sumsets, Set addition, Khovanskii polynomial, Structure Theorem, Explicit Bound