Explicit sumset sizes in additive number theory
Melvyn B. Nathanson
TL;DR
The paper investigates the full range of sumset sizes $\mathcal{R}_{\mathbf Z}(h,k)$ for finite sets of integers, focusing on $h\ge3$ and $k\ge3$, a problem with known bounds but many missing values. It develops two explicit constructive frameworks: (a) sums of arithmetic progressions and (b) sums of intervals of different lengths, to produce infinite families of sets with prescribed $|hA|$ and to locate elements of $\mathcal{R}_{\mathbf Z}(h,k)$. It shows that $\mathcal{R}_{\mathbf Z}(h,k)$ is not an interval for $h\ge3,k\ge3$, providing specific missing numbers such as $hk-h+2$, and identifies large contiguous subranges and arithmetic progressions inside $\mathcal{R}_{\mathbf Z}(h,k)$, including $hk\in\mathcal{R}_{\mathbf Z}(h,k)$ and $(h+1)^2\in\mathcal{R}_{\mathbf Z}(h,4)$. The results contribute explicit constructions and interval-like lower bounds that help map the landscape of sumset sizes and guide conjectures.
Abstract
It is an open problem in additive number theory to compute and understand the full range of sumset sizes of finite sets of integers, that is, the set $\mathcal{R}_{\mathbf{Z}}(h,k)= \{|hA|:A \subseteq {\mathbf{Z}} \text{ and } |A|=k\}$ for all integers $h \geq 3$ and $k \geq 3$. This paper constructs certain infinite families of finite sets of size $k$ and computes their $h$-fold sumset sizes.