Sum-dominant sets and restricted-sum-dominant sets in finite abelian groups
Volume 165 / 2014
Acta Arithmetica 165 (2014), 361-383
MSC: Primary 11B75.
DOI: 10.4064/aa165-4-6
Abstract
We call a subset $A$ of an abelian group $G$ sum-dominant when $\def\abs#1{\vert#1\vert}\abs{A+A}>\abs{A-A}$. If $\def\abs#1{\vert#1\vert}\abs{A\mathbin{\hat{+}}A}>\abs{A-A}$, where $A\mathbin{\hat{+}}A$ comprises the sums of distinct elements of $A$, we say $A$ is restricted-sum-dominant. In this paper we classify the finite abelian groups according to whether or not they contain sum-dominant sets (respectively restricted-sum-dominant sets). We also consider how much larger the sumset can be than the difference set in this context. Finally, generalising work of Zhao, we provide asymptotic estimates of the number of restricted-sum-dominant sets in finite abelian groups under mild conditions.
