On Sums of Generating Sets in (Z_2)^n

Details On Sums of Generating Sets in (Z_2)^n On Sums of Generating Sets in (Z_2)^n

2011-08-24

arxiv.org/abs/1108.4902v1

Mathematics

Combinatorics

21 pages, 3 illustrations

Scientific paper

Let A and B be two affinely generating sets of (Z_2)^n. As usual, we denote their Minkowski sum by A+B. How small can A+B be, given the cardinalities of A and B? We give a fairly tight answer to this question. Our bound is attained when both A and B are unions of cosets of a certain subgroup of (Z_2)^n. These cosets are arranged as Hamming balls, the smaller of which has radius 1. By similar methods, we reprove the Freiman-Ruzsa theorem in (Z_2)^n, with an optimal upper bound. Denote by F(K) the maximal spanning constant || / |A|, over all subsets A of (Z_2)^n with doubling constant |A+A| / |A| < K. We explicitly calculate F(K), and in particular show that 4^K / 4K < F(K) (1+o(1)) < 4^K / 2K. This improves the estimate F(K) = poly(K) 4^K, found recently by Green and Tao and by Konyagin.

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