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Restricted sumsets in finite nilpotent groups

Volume 178 / 2017

Shanshan Du, Hao Pan Acta Arithmetica 178 (2017), 101-123 MSC: Primary 11P70; Secondary 11B13. DOI: 10.4064/aa7437-8-2016 Published online: 23 March 2017

Abstract

Suppose that $A,B$ are non-empty subsets of the finite nilpotent group $G$. If $A\not=B$, then the cardinality of the restricted sumset $$A\mathbin\dotplus B=\{a+b:a\in A,\, b\in B,\, a\neq b\} $$ is at least $$\min\{p(G),|A|+|B|-2\},$$ where $p(G)$ denotes the least prime factor of $|G|$. Moreover we prove that if $A$ is a non-empty subset of a finite group $G$ with $|A| \lt (p(G)+3)/2$, then the elements of $A$ commute when $$ |A\mathbin\dotplus A|=2|A|-3. $$

Authors

  • Shanshan DuThe Fundamental Division
    Jingling Institute of Technology
    Nanjing 211169
    People’s Republic of China
    e-mail
  • Hao PanDepartment of Mathematics
    Nanjing University
    Nanjing 210093
    People’s Republic of China
    e-mail

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