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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 differences of elements of A, respectively. The well-known inequality $σ(A)^{1/2} \leq δ(A) \leq σ(A)^2,$ where $σ(A)=\frac{|A+A|}{|A|}$ is the doubling constant of A and $δ(A)=\frac{|A-A|}{|A|}$ is the difference constant of A, relates the relative sizes of the sumset and difference set of A. The exponent 2 in this inequality is known to be optimal, for the exponent 1/2 this is unknown. We determine those sets for which equality holds in the above inequality. 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 difference constant are equal to 1. This implies that there is space for possible improvement of the exponent 1/2 in the inequality. We then use the derived methods to show that Plünnecke's inequality is strict when the doubling constant is larger than 1.