The restricted sumsets in finite abelian groups

Abstract

Suppose that $k\ge 2$ and $A$ is a non-empty subset of a finite abelian group $G$ with $|G|>1$. Then the cardinality of the restricted sumset $$k^{\wedge }A:=\{a_1+\cdots +a_k:a_1,\dots ,a_k\in A,a_i\ne a_j\text{ for }i\ne j\}$$ is at least $$\min \{p(G),k|A|-k^2+1\},$$ where $p(G)$ denotes the least prime divisor of $|G|$.

9 pages. Accepted by the Israel Journal of Mathematics