The relative sizes of sumsets and difference sets

Abstract

Let A be a finite subset of a commutative additive group Z. The sumset and difference set of A are defined as the sets of pairwise sums and di↵erences of elements of A, respectively. The well-known inequality (A)1/2  (A)  (A)2, where (A) = |A+A| |A| is the doubling constant of A and (A) = |AA| |A| is the di↵erence constant of A, relates the relative sizes of the sumset and di↵erence set of A. The exponent 2 in this inequality is known to be optimal. For the exponent 1 2 this is unknown. Here, we determine the equality case of both inequalities. For both inequalities we find that equality holds if and only if A is a coset of some finite subgroup of Z or, equivalently, if and only if both the doubling constant and di↵erence constant are equal to 1. This is a necessary condition for possible improvement of the exponent 1 2 . We then use the derived methods to show that Plunnec ¨ ke’s inequality is strict when the doubling constant is larger than 1.