The relative sizes of sumsets and difference sets

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.