On the generalized restricted sumsets in abelian groups

TL;DR

This paper establishes lower bounds on the size of generalized restricted sumsets in finite abelian groups, extending classical sumset results by incorporating restrictions based on a subset S.

Contribution

It introduces new bounds for restricted sumsets in abelian groups, generalizing existing sumset theorems with restrictions involving a subset S.

Findings

Lower bound: |A S+B| ≥ min{|A|+|B|-3|S|, p(G)}

Improved bound for large |A|, |B|: |A S+B| ≥ min{|A|+|B|-|S|-2, p(G)}

Results depend on the size of subsets and the prime factor p(G).

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 $|S|$.