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

Jiantao Li, Xinyi Liang

Let $p$ be a prime and let $A,B$ be nonempty subsets of the cyclic group ${\mathbb{Z}}_p$ with $|A|\ne |B|$. The Alon--Nathanson--Ruzsa theorem gives the lower bound $|A\rplus B|\ge…