Restricted Sumsets in Finite Nilpotent Groups

Abstract

Suppose that $A,B$ are two non-empty subsets of the finite nilpotent group $G$. If $A\ne B$, then the cardinality of the restricted sumset $$A\dotplus B=a+b:a\in A,b\in B,a\ne b$$ is at least $$\min p(G),|A|+|B|-2,$$ where $p(G)$ denotes the least prime factor of $|G|$.

This is a preliminary draft, which maybe contains some mistakes. Now Theorem 1.2 has been extended to general finite groups