Sets with few differences in abelian groups

2015/08/22 by Lee, Mitchell
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1508.05524

Abstract

Let (G, +) be an abelian group. In 2004, Eliahou and Kervaire found an explicit formula for the smallest possible cardinality of the sumset A+A, where A ⊆ G has fixed cardinality r. We consider instead the smallest possible cardinality of the difference set A-A, which is always greater than or equal to the smallest possible cardinality of A+A and can be strictly greater. We conjecture a formula for this quantity and prove the conjecture in the case that G is a cyclic group or a vector space over a finite field. This resolves a conjecture of Bajnok and Matzke on signed sumsets.

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• Problems in additive number theory, VI: Sizes of sumsets
• Difference sets and positive exponential sums I. General properties