Critical Pairs for Mixed Restricted Sumsets in Prime Fields

Abstract

Let $p$ be an odd prime and let $A,B\subseteq {\mathbb{F}}_p$ be nonempty and distinct, with $|A|\ge |B|$. We give a complete classification of the pairs for which the mixed restricted sumset $A\widehat{+}B=\{x+y:x\in A,y\in B,x\ne y\}$ has size $\min \{p,|A|+|B|-2\}$. The main new contribution concerns the interior range $|A|+|B|\le p-1$. For $|B|\ge 3$ and $|A|-|B|\ge 3$, criticality forces $A$ and $B$ to be common-endpoint arithmetic progressions, apart from a single common affine orbit with $(p,|A|,|B|)=(13,7,4)$. For $|A|-|B|\in \{1,2\}$, criticality itself forces $B⊊A$, after which the restricted self-sum determines the deletion patterns. Together with the remaining cases, this yields the complete classification.