TITLE: Large sets $X \subseteq \mathbb{Z}_2^n$ with $X+X \ne \mathbb{Z}_2^n$
QUESTION [11 upvotes]: I've come across a seemingly natural question in additive combinatorics to which I can't find the answer in the literature. I would like to know if, given an $\alpha > 0$ and large $n$, there are interesting examples of sets $X \subseteq \mathbb{Z}_2^n$ such that $|X| \ge \alpha 2^n$ and $X + X \ne \mathbb{Z_2}^n$. 
There are certainly uninteresting examples: say, the set of points in some suitably large proper subspace of $\mathbb{Z_2}^n$, or some union of cosets of such a subspace, etc etc. I would like to know if there are examples that do not have an obvious linear-algebraic structure of this sort. specifically, are there examples that are 'pseudorandom', say such that every codimension-1 subspace contains roughly $|X|/2$ elements of $X$? 
In the language of additive theory, we know that $X$ has 'doubling constant' $\frac{|X+X|}{|X|} < \alpha^{-1}$, and there are many results in additive theory describing sets with small doubling constant.  However, the same inequality also holds for an arbitrary set strictly larger than $X$, so no result on doubling constant alone will give sensible structure. Is there a well-known theorem or observation somewhere that I'm missing?

REPLY [2 votes]: As a slight extension of my comment, one can fix a (very) small subset $S\subset{\mathbb Z}_2^n$, and consider the addition Cayley graph, say $\Gamma$, induced by $S$ on ${\mathbb Z}_2^n$. (The vertices of $\Gamma$ are the elements of ${\mathbb Z}_2^n$, with two vertices adjacent whenever their sum is in $S$.) Now form an independent set $X\subset {\mathbb Z}_2^n$ by choosing at random, for every edge of $\Gamma$, one of the two vertices incident with this edge, and deleting this vertex from ${\mathbb Z}_2^n$. By the construction, we have $2X\subseteq{\mathbb Z}_2^n\setminus S$; furthermore, the expected density of $X$ is $2^{-|S|}$ and, typically, $X$ will be very uniformly distributed in ${\mathbb Z}_2^n$.