On two questions about restricted sumsets in finite abelian groups

TL;DR

This paper studies the maximum sizes of special subsets in finite abelian groups that are incomplete or zero-sum-free with respect to sums of a fixed number of elements, providing exact values for certain group types and parameters.

Contribution

It determines the maximum sizes of weakly h-incomplete and weakly h-zero-sum-free sets in finite abelian groups for specific cases, extending understanding of sumset restrictions.

Findings

For odd order groups with certain h, the maximum size is h+1.

For elementary abelian 2-groups with certain h, the maximum size is h+2.

Exact values are given except for specific small cases.

Abstract

Let $G$ be an abelian group of finite order $n$, and let $h$ be a positive integer. A subset $A$ of $G$ is called {\em weakly $h$-incomplete}, if not every element of $G$ can be written as the sum of $h$ distinct elements of $A$; in particular, if $A$ does not contain $h$ distinct elements that add to zero, then $A$ is called {\em weakly $h$-zero-sum-free}. We investigate the maximum size of weakly $h$-incomplete and weakly $h$-zero-sum-free sets in $G$, denoted by $C_h(G)$ and $Z_h(G)$, respectively. Among our results are the following: (i) If $G$ is of odd order and $(n-1)/2 \leq h \leq n-2$, then $C_h(G)=Z_h(G)=h+1$, unless $G$ is an elementary abelian 3-group and $h=n-3$; (ii) If $G$ is an elementary abelian 2-group and $n/2 \leq h \leq n-2$, then $C_h(G)=Z_h(G)=h+2$, unless $h=n-4$.