On sumsets of nonbases of maximum size

TL;DR

This paper investigates the possible sizes of sumsets of nonbases of maximum size in finite abelian groups, providing a complete characterization for cases when the order is 2 or 3.

Contribution

It offers a complete classification of sumset sizes for nonbases of maximum size in finite abelian groups for orders 2 and 3.

Findings

Characterization of sumset sizes for nonbases of order 2

Characterization of sumset sizes for nonbases of order 3

Complete solutions for maximum size nonbases in finite abelian groups

Abstract

Let $G$ be a finite abelian group. A nonempty subset $A$ in $G$ is called a basis of order $h$ if $hA=G$; when $hA \neq G$, it is called a nonbasis of order $h$. Our interest is in all possible sizes of $hA$ when $A$ is a nonbasis of order $h$ in $G$ of maximum size; we provide the complete answer when $h=2$ or $h=3$.