Restricted sumsets and a conjecture of Lev

Abstract

Let A,B,S be finite subsets of an abelian group G. Suppose that the restricted sumset C={a+b: a in A, b in B, and a-b not in S} is nonempty and some c in C can be written as a+b with a in A and b in B in at most m ways. We show that if G is torsion-free or elementary abelian then |C|\geq |A|+|B|-|S| -m. We also prove that |C|\geq |A|+|B|-2|S|-m if the torsion subgroup of G is cyclic. In the case S={0} this provides an advance on a conjecture of Lev.