On the Minimum Size of Signed Sumsets in Elementary Abelian Groups

TL;DR

This paper investigates the minimal size of signed sumsets in elementary abelian groups, providing exact values and conditions, especially for groups of the form rac{rac{ p^2}{m}{2} and comparing with non-signed sumsets.

Contribution

It determines when the minimal sizes of signed and unsigned sumsets are equal in elementary abelian groups, extending previous bounds and results.

Findings

Exact values of rac{rac{ p^2}{m}{2} for which rac{rac{ p^2}{m}{2} equals the unsigned sumset size.

Conditions under which signed and unsigned sumsets have the same minimal size in elementary abelian groups.

Extension of previous bounds to specific classes of elementary abelian groups.

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 $h$. In previous work we provided an upper bound for $\rho_{\pm} (G, m, h)$ that we believe is exact, and proved that $\rho_{\pm} (G, m, h)$ agrees with $\rho (G, m, h)$ when $G$ is cyclic. Here we study $\rho_{\pm} (G, m, h)$ for elementary abelian groups $G$; in particular, we determine all values of $m$ for which $\rho_{\pm} (\mathbb{Z}_p^2, m, 2)$ equals $\rho (\mathbb{Z}_p^2, m, 2)$ for a given prime $p$.