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 $$AB:=\{a+b:a\in A,b\in B,a-b\in ̸S\}$$ contains at least $$\min \{\mathrm{\mid }A\mathrm{\mid }+\mathrm{\mid }B\mathrm{\mid }-3\mathrm{\mid }S\mathrm{\mid },p(G)\}$$ elements, where $p(G)$ is the least prime factor of $\mathrm{\mid }G\mathrm{\mid }$. Further, we also have $$\mathrm{\mid }AB\mathrm{\mid }\ge \min \{\mathrm{\mid }A\mathrm{\mid }+\mathrm{\mid }B\mathrm{\mid }-\mathrm{\mid }S\mathrm{\mid }-2,p(G)\},$$ provided that both $\mathrm{\mid }A\mathrm{\mid }$ and $\mathrm{\mid }B\mathrm{\mid }$ are large with respect to $\mathrm{\mid }S\mathrm{\mid }$.