Critical Restricted Sumsets at the Boundary $\lvert A\rvert+\lvert B\rvert=p$

Abstract

Let $p$ be an odd prime, and write $A\RS B=\{a+b:a\in A,\ b\in B,\ a\ne b\}$. We classify all pairs $A,B\subseteq\Fp$ satisfying $|A|+|B|=p$, $|A|>|B|=\ell\ge1$, and $|A\RS B|=p-2$. After normalizing the two missing sums to $\{0,1\}$, we obtain exactly $\ell+1$ explicit models. The classification forces $B\subseteq A$ and yields, for fixed $p,\ell$, exactly $\binom p2(\ell+1)$ ordered pairs and $1+\lfloor\ell/2\rfloor$ equivalence classes under simultaneous affine transformations. We determine when either set is an arithmetic progression and compute exact difference-set cardinalities. For $\ell\ge3$ and $p\ge4\ell-5$, the cardinality $|B-B|$ determines the equivalence class at fixed $p,\ell$. The proof uses punctured translates and cyclic component counting. We also exhibit a critical pair in $\mathbb F_{13}$, with size gap three and $|A|+|B|=p-2$, that contradicts a proposed inverse statement below the boundary.