A density version of the Carlson--Simpson theorem

We prove a density version of the Carlson--Simpson Theorem. Specifically we show the following. For every integer $k\geq 2$ and every set $A$ of words over $k$ satisfying \[\limsup_{n\to\infty} \frac{|A\cap [k]^n|}{k^n}>0\] there exist a word $c$ over $k$ and a sequence $(w_n)$ of left variable words over $k$ such that the set \[\{c\}\cup \big\{c^{\smallfrown}w_0(a_0)^{\smallfrown}...^{\smallfrown}w_n(a_n) : n\in\mathbb{N} \ \text{ and } \ a_0,...,a_n\in [k]\big\}\] is contained in $A$. While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.