On Sums of Generating Sets in ℤ2n

Abstract

LetAandBbe two affinely generating sets of ℤ2n. As usual, we denote their Minkowski sum byA+B. How small canA+Bbe, given the cardinalities of A andB? We give a tight answer to this question. Our bound is attained when bothAandBare unions of cosets of a certain subgroup of ℤ2n. These cosets are arranged as Hamming balls, the smaller of which has radius 1.By similar methods, we re-prove the Freiman–Ruzsa theorem in ℤ2n, with an optimal upper bound. Denote byF(K)the maximal spanning constant |〈A〉|/|A| over all subsetsA⊆ ℤ2nwith doubling constant |A+A|/|A| ≤K. We explicitly calculateF(K), and in particular show that 4K/4K≤F(K)⋅(1+o(1)) ≤ 4K/2K. This improves the estimateF(K)= poly(K)4K, found recently by Green and Tao [17] and by Konyagin [23].