On the generalized restricted sumsets in abelian groups

Abstract

Suppose that $A$, $B$ and $S$ are non-empty subsets of a finite abelian group $G$. Then the generalized restricted sumset $$A\stackrel{S}{+}B:=\{a+b:a\in A,b\in B,a-b\notin S\}$$ contains at least $$\min \{|A|+|B|-3|S|,p(G)\}$$ elements, where $p(G)$ is the least prime factor of $|G|$. Further, we also have $$|A\stackrel{S}{+}B|\ge \min \{|A|+|B|-|S|-2,p(G)\},$$ provided that both $|A|$ and $|B|$ are large with respect to $|S|$.

This is a very preliminary draft