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

Abstract

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

20 pages, submitted., Topic of a talk given at QTMC2025