Restricted set addition in finite abelian groups

Abstract: Let $A$ be a nonempty subset of a 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 $|A| > \frac{n}{2}$, then $h^{\wedge}A = G$ under some conditions on $h$ and $n$. While the constant $1/2$ is optimal for groups of even order, it is not optimal 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 talk, we discuss 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 when the smallest prime dividing $n$ is $3$.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)