On the generalized restricted sumsets in abelian groups

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 $|S|$.

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