Covering $\mathbb{F}_2^n$ with Hamming Balls

Abstract

Green asked the following question concerning structures in sumsets: Suppose that $\mathbb{F}_2^n$ is partitioned into sets $A_1, \dots, A_K$. Does $A_i+A_i$ contain a coset of codimension $O_K(1)$ for some $i$? An answer is not known even in the case of $K = 3$. We resolve this question in the affirmative in two special cases: (1) when $A_1, A_2$ are Hamming balls of radius $r < n/2 - 7$ relative to different bases, and (2) when $A_1$ is a Hamming ball of sufficiently small constant density.