On Sums of Generating Sets in (Z_2)^n

TL;DR

This paper determines the minimal size of the sumset of two affinely generating sets in (Z_2)^n, proves a version of the Freiman-Ruzsa theorem with optimal bounds, and explicitly calculates the maximal spanning constant related to doubling constants.

Contribution

It provides a tight bound for the sumset size of affinely generating sets in (Z_2)^n and re-proves the Freiman-Ruzsa theorem with optimal bounds, improving previous estimates.

Findings

The minimal sumset size is attained when A and B are unions of cosets arranged as Hamming balls.

Explicit calculation of the maximal spanning constant F(K) for subsets with doubling constant less than K.

Improved bounds on F(K), showing it grows roughly as 4^K / (K).

Abstract

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 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 re-prove the Freiman-Ruzsa theorem in (Z_2)^n, with an optimal upper bound. Denote by F(K) the maximal spanning constant |<A>|/|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.