Large restricted sumsets in general abelian group

TL;DR

This paper investigates the size and structure of restricted sumsets in finite abelian groups, establishing bounds and characterizations for when these sumsets cover the entire group or are close to it.

Contribution

It introduces new bounds and structural characterizations for restricted sumsets in finite abelian groups, extending classical sumset results to more general settings.

Findings

If |A|+|B|>|G|+L_S then A∧^S B=G

When |A|+|B|=|G|+L_S, then |A∧^S B|>=|G|-2|S|

Characterization of (A,B,S) triples achieving equality in bounds

Abstract

Let A, B and S be three subsets of a finite Abelian group G. The restricted sumset of A and B with respect to S is defined as A\wedge^{S} B= {a+b: a in A, b in B and a-b not in S}. Let L_S=max_{z in G}| {(x,y): x,y in G, x+y=z and x-y in S}|. A simple application of the pigeonhole principle shows that |A|+|B|>|G|+L_S implies A\wedge^S B=G. We then prove that if |A|+|B|=|G|+L_S then |A\wedge^S B|>= |G|-2|S|. We also characterize the triples of sets (A,B,S) such that |A|+|B|=|G|+L_S and |A\wedge^S B|= |G|-2|S|. Moreover, in this case, we also provide the structure of the set G\setminus (A\wedge^S B).