A result on the size of iterated sumsets in $\mathbb{Z}^d$

TL;DR

This paper introduces a new approach to analyze the size of iterated sumsets in integer lattices, providing simplified proofs and bounds for specific set sizes.

Contribution

It offers a novel method for determining sumset cardinalities and extends results to larger sets with simplicial hulls in .

Findings

Derived bounds for  sumsets with d+2 elements

Provided an upper bound for sumsets of d+3 elements with simplicial hull

Simplified proof techniques for sumset size determination

Abstract

In this paper we give a different approach to determining the cardinality of $h$-fold sumsets $hA$ when $A\subset \mathbb{Z}^d$ has $d+2$ elements. This enables us to provide more general result with a shorter and simpler proof. We also obtain an upper bound for the value of $|hA|$ when $A\subset \mathbb{Z}^d$ is a set of $d+3$ elements with simplicial hull.