Inverse problem for critical restricted sumset pairs in Z_p

The problem asks for an inverse classification of critical restricted sumset pairs in a prime cyclic group. The latest report says the conjecture fails when $|A|+|B|=p$, but a corrected result holds when $|A|+|B|\leq p-1$.

Known results

For $A,B\subseteq\mathbb{Z}/p\mathbb{Z}$ with $|A|=|B|=k$, a 2024 paper claims that $|A\dot{+}B|=2k-2$ implies $A=B$ for $k\geq 5$ and $p>2k-2$; Károlyi’s earlier work characterized the preceding equality case, with stated exceptions.

September 2026 counterexample

A report dated September 9, 2026 identifies a counterexample to the Liu–Qian conjecture at $|A|+|B|=p$ and states a refined inverse theorem under $|A|+|B|\leq p-1$. The unrestricted inverse problem therefore remains open.

Current status (as of September 2026): the boundary case $|A|+|B|=p$ is claimed to be false and the restricted range $|A|+|B|\leq p-1$ is claimed to be settled, but the unrestricted inverse problem remains open.