Inverse problems for sumset sizes of finite sets of integers

TL;DR

This paper explores the properties and relationships of sumset sizes of finite integer sets, focusing on their growth, configurations, and differences for affinely inequivalent sets.

Contribution

It introduces new insights into the behavior and comparison of sumset size sequences for finite integer sets, including their growth rates and structural differences.

Findings

Characterization of sumset size sequences

Relations between sumset sequences of inequivalent sets

Analysis of growth rates and configurations

Abstract

Let $A$ be a finite set of integers and let $hA$ be its $h$-fold sumset. This paper investigates the sequence of sumset sizes $( |hA| )_{h=1}^{\infty}$, the relations between these sequences for affinely inequivalent sets $A$ and $B$, and the comparative growth rates and configurations of the sumset size sequences $( |hA| )_{h=1}^{\infty}$ and $( |hA| )_{h=1}^{\infty}$.