Restricted sums of sets of cardinality $2p + 1$ in $\mathbb{Z}_p^2$

TL;DR

This paper proves a lower bound on the size of the restricted sumset of a set with size 2p+1 in a0a0p^2, advancing the understanding of sumset cardinalities in finite abelian groups.

Contribution

It establishes the first significant lower bound for the restricted sumset of such sets in over twenty years, addressing a variant of the Erdf6s-Heilbronn problem.

Findings

The restricted sumset has size at least 4p.

First progress in over two decades on this problem.

Provides new bounds for sumsets in a0a0p^2.

Abstract

Let $A\subseteq \mathbb{Z}_p^2$ be a set of size $2p+1$ for prime $p\geq 5$. In this paper, we prove that $A\hat{+}A=\{a_1+a_2\mid a_1,a_2\in A, a_1\neq a_2\}$ has cardinality at least $4p$. This result is the first advancement in over two decades on a variant of the Erd\H{o}s-Heilbronn problem studied by Eliahou and Kervaire.