Open problems about sumsets in finite abelian groups: minimum sizes and critical numbers

TL;DR

This paper reviews known and unknown results about the minimum sizes and critical numbers of sumsets, restricted sumsets, and signed sumsets in finite abelian groups, highlighting new open problems in the area.

Contribution

It provides a comprehensive review of existing results and introduces new open problems related to sumset sizes and critical numbers in finite abelian groups.

Findings

Summary of known bounds for sumset sizes

Identification of gaps in current understanding

Proposals for future research directions

Abstract

For a positive integer $h$ and a subset $A$ of a given finite abelian group, we let $hA$, $h \hat{\;} A$, and $h_{\pm}A$ denote the $h$-fold sumset, restricted sumset, and signed sumset of $A$, respectively. Here we review some of what is known and not yet known about the minimum sizes of these three types of sumsets, as well as their corresponding critical numbers. In particular, we discuss several new open direct and inverse problems.