Green's sumset problem at density one half

TL;DR

This paper explores the structure of sumsets in binary vector spaces, demonstrating that high-density subsets have sumsets containing large subspaces with specific co-dimensions.

Contribution

It establishes new bounds on the co-dimension of subspaces contained in sumsets of dense subsets in F_2^n, advancing understanding of Green's sumset problem.

Findings

Sumsets of subsets with density ≥ 1/2 - o(n^{-1/2}) contain a co-dimension 1 subspace.

Sumsets of subsets with density ≥ 1/2 - o(1) contain a subspace of co-dimension o(n).

Provides bounds on the structure of sumsets at high density levels.

Abstract

We investigate the size of subspaces in sumsets and show two main results. First, if A is a subset of F_2^n with density at least 1/2 - o(n^{-1/2}) then A+A contains a subspace of co-dimension 1. Secondly, if A is a subset of F_2^n with density at least 1/2-o(1) then A+A contains a subspace of co-dimension o(n).