On a missing interval of integers from $\mathcal{R}_{\mathbf{Z}}(h,4)$

Abstract: The set $\mathcal{R}_{\Z}(h,4)$ consists of all possible sizes for the $h$-fold sumset of sets containing four integers. An immediate question to ask is what are the elements of this set? We know that $\mathcal{R}_{\Z}(h,4)\subseteq [3h+1,\binom{h+3}{h}]$, where the right side is an interval of integers that includes the endpoints. These endpoints are known to be attained. By observation, it appears that the interval of integers $[3h+2,4h-1]$ is absent from $\mathcal{R}_{\Z}(h,4)$. We will briefly discuss the procedure used to prove that the integers in $[3h+2,4h-1]$ are not possible sizes for the $h$-fold sumset of a set containing four integers. Furthermore, we will confirm that this interval can't be made larger by exhibiting a set whose h-fold sumset has size $4h$.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)