Critical Restricted Sumsets at the Boundary $|A|+|B|=p$

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 $|A|+|B|=p$, $|A|>|B|=\ell \ge 1$, 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 $\left(\genfrac{}{}{0ex}{}{p}{2}\right)(\ell +1)$ 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 $\ell \ge 3$ and $p\ge 4\ell -5$, 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.

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