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Sum-dominant sets and restricted-sum-dominant sets in finite abelian groups

Volume 165 / 2014

David B. Penman, Matthew D. Wells 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.

Authors

  • David B. PenmanDepartment of Mathematical Sciences
    University of Essex
    Wivenhoe Park
    Colchester CO4 3SQ, United Kingdom
    e-mail
  • Matthew D. WellsDepartment of Mathematical Sciences
    University of Essex
    Wivenhoe Park
    Colchester CO4 3SQ, United Kingdom
    e-mail

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