On the Sumset of Sets of Size $k$

TL;DR

This paper investigates the possible sizes of h-fold sumsets of k-element subsets in additive groups, establishing new restrictions on achievable sumset sizes and confirming certain maximal sizes are attainable.

Contribution

It proves that certain intermediate sumset sizes are impossible for sets with at least four elements in torsion-free abelian groups, refining the understanding of sumset size ranges.

Findings

Integers in [hk-h+2, hk-1] cannot be sumset sizes for k≥4.

The endpoints of the sumset size interval are attainable.

The maximum sumset size hk is achievable with specific subsets.

Abstract

The set $\mathcal{R}_{G}(h,k)$ consists of all possible sizes for the $h$-fold sumset of sets containing $k$ elements from an additive abelian group $G$. The exact makeup of this set is still unknown, but there has been progress towards determining which integers are present. We know that $\mathcal{R}_{G}(h,k)\subseteq\left[hk-h+1,\binom{h+k-1}{h}\right]$, where the right side is an interval of integers that includes the endpoints. These endpoints are known to be attained. We will prove that the integers in $\left[hk-h+2,hk-1\right]$ are not possible sizes for the $h$-fold sumset of a set containing $k\geq 4$ elements of a torsion-free additive abelian group $G$. Furthermore, we will confirm that this interval can't be made larger by exhibiting a subset of $G$ whose $h$-fold sumset has size $hk$.