Critical Pairs for Mixed Restricted Sumsets in Prime Fields

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