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 $h \geq 2$, 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 $h \geq 4$, let $α_h$ be the unique positive root of the polynomial $3^{h - 2} x^{h - 1} + x - 1$. In this paper, we show that for any $α> α_h$, there exists a positive integer $M_h(α)$, which is determined precisely, such that for all $n > M_h(α)$ with $n$ odd, if $A$ is a subset of a finite abelian group $G$ of order $n$ and if $|A| \geq αn$, then $h^{\wedge} A = G$. Moreover, $α_h > α_{h + 1}$ for $h \geq 4$ and $α_h$ 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 $h \geq 4$, and to arbitrary finite abelian groups.