A Counterexample to a Conjecture of Liu and Qian and a Refined Inverse Theorem for Restricted Sumsets in $\mathbb{Z}_p$

Abstract: Let $p$ be a prime and let $A,B$ be nonempty subsets of the cyclic group $\mathbb{Z}_p$ with $|A|\neq |B|$. The Alon--Nathanson--Ruzsa theorem gives the lower bound [ |A\rplus B|\ge \min{p,\,|A|+|B|-2}, ] where $A\rplus B={a+b:a\in A,\ b\in B,\ a\neq b}$ is the restricted sumset. The inverse problem of characterizing all critical pairs $(A,B)$ attaining equality was posed by Alon, Nathanson, and Ruzsa in 1996 and remains open. Recently, Liu and Qian solved the inverse problem under the assumption that at least one of the sets is an arithmetic progression, and proposed a conjecture for the general case. In this paper, we first exhibit a counterexample to their conjecture, which arises in the boundary case $|A|+|B|=p$. This counterexample shows that the original conjecture is false without additional restrictions. Motivated by this, we formulate and prove a refined inverse theorem under the natural hypothesis $|A|+|B|\le p-1$. Our proof relies on the results of Liu and Qian together with a new counting argument.