Restricted sumsets in finite nilpotent groups
Volume 178 / 2017
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 $$ |A\mathbin\dotplus A|=2|A|-3. $$
