The Minimum Size of Signed Sumsets

TL;DR

This paper investigates the minimal size of signed sumsets in finite abelian groups, proving equality with sumsets in cyclic groups and proposing a conjectured exact value for all groups.

Contribution

It establishes that the minimal signed sumset size equals the sumset size in cyclic groups and provides an upper bound conjectured to be exact for all finite abelian groups.

Findings

quality of  ho_{\u00b1}(G, m, h) and ho(G, m, h) in cyclic groups

n upper bound for  ho_{\u00b1}(G, m, h) for all finite abelian groups

vidence suggesting the upper bound is the exact value

Abstract

For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$\rho(G, m, h) = \min \{|hA| \; : \; A \subseteq G, |A|=m\}$$ and $$\rho_{\pm} (G, m, h) = \min \{|h_{\pm} A| \; : \; A \subseteq G, |A|=m\},$$ where $hA$ and $h_{\pm} A$ denote the $h$-fold sumset and the $h$-fold signed sumset of $A$, respectively. The study of $\rho(G, m, h)$ has a 200-year-old history and is now known for all $G$, $m$, and $h$. Here we prove that $\rho_{\pm}(G, m, h)$ equals $\rho (G, m, h)$ when $G$ is cyclic, and establish an upper bound for $\rho_{\pm} (G, m, h)$ that we believe gives the exact value for all $G$, $m$, and $h$.