Restricted set addition in finite abelian groups

TL;DR

This paper establishes conditions under which the restricted h-fold sumset of a large subset in a finite abelian group covers the entire group, generalizing previous results from cyclic groups to all finite abelian groups.

Contribution

It introduces a precise threshold for subset size ensuring the sumset equals the group, extending known cyclic group results to all finite abelian groups.

Findings

For any  > _h, large enough groups have sumsets covering the entire group.

The thresholds _h decrease with h and approach 1/3, which is proven optimal.

The results generalize prior cyclic group findings to all 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 $\alpha_h$ be the unique positive root of the polynomial $3^{h - 2} x^{h - 1} + x - 1$. In this paper, we show that for any $\alpha > \alpha_h$, there exists a positive integer $M_h(\alpha)$, which is determined precisely, such that for all $n > M_h(\alpha)$ with $n$ odd, if $A$ is a subset of a finite abelian group $G$ of order $n$ and if $|A| \geq \alpha n$, then $h^{\wedge} A = G$. Moreover, $\alpha_h > \alpha_{h + 1}$ for $h \geq 4$ and $\alpha_h$ approaches $\frac{1}{3}$ as $h$ increases, and the constant $\frac{1}{3}$ is optimal. This result generalizes a result of Tang and Wei for $4^{\wedge}A$ in the cyclic group $\mathbb{Z}_n$ to arbitrary finite abelian groups.