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 $$ A\stackrel{S}+B:={a+b:\,a\in A,\ b\in B,\ a-b\not\in S} $$ 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 $$ |A\stackrel{S}+B|\geq \min{|A|+|B|-|S|-2,p(G)}, $$ provided that both $|A|$ and $|B|$ are large with respect to $|S|$.