Explicit sumset sizes in additive number theory

TL;DR

This paper investigates the full range of sumset sizes for finite integer sets, constructing specific families to compute their h-fold sumset sizes, advancing understanding in additive number theory.

Contribution

It introduces new infinite families of finite sets and explicitly computes their sumset sizes, addressing an open problem in additive number theory.

Findings

Constructed infinite families of finite sets with known sumset sizes

Computed h-fold sumset sizes for these families

Contributed to understanding the range of sumset sizes

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.