Restricted set addition in finite abelian groups

Abstract

Let $A$ be a nonempty subset of finite abelian group $G$ of order $n$. For an integer $h\ge 2$, the restricted $h$-fold sumset $h^{\wedge }A$ is the set of all sums of $h$ distinct elements of $A$. It is known that if $G$ is a group of order $n$ and $A$ is a subset of $G$ such that $|A|$ is close to $\frac{n}{2}$, then $h^{\wedge }A=G$ under some conditions on $h$ and $n$. The constant $\frac{1}{2}$ is optimal for groups of even order but not for groups of odd order. For an integer $h\ge 4$, let $\hat{I}{\pm }_h$ be the unique positive root of the polynomial $3^{h-2}{x}^{h-1}+x-1$. In this paper, we show that for any $\hat{I}\pm >\hat{I}{\pm }_h$, there exists a positive integer $M_h(\hat{I}\pm )$, which is determined precisely, such that for all $n>M_h(\hat{I}\pm )$ with $n$ odd, if $A$ is a subset of a finite abelian group $G$ of order $n$ and if $|A|\ge \hat{I}\pm n$, then $h^{\wedge }A=G$. Moreover, $\hat{I}{\pm }_h>\hat{I}{\pm }_{h+1}$ for $h\ge 4$ and $\hat{I}{\pm }_h$ approaches $\frac{1}{3}$ as $h$ increases, and the constant $\frac{1}{3}$ is optimal when the smallest prime dividing $n$ is $3$. This result extends a theorem of Tang and Wei on $4^{\wedge }A$ in the cyclic group ${\mathbb{Z}}_n$ to $h^{\wedge }A$ for every $h\ge 4$, and to arbitrary finite abelian groups.

18 pages