Measurable events indexed by words

Abstract

For every integer $k\geq 2$ let $[k]^{<\mathbb{N}}$ be the set of all words over $k$, that is, all finite sequences having values in $[k]:=\{1,...,k\}$. A Carlson-Simpson tree of $[k]^{<\mathbb{N}}$ of dimension $m\geq 1$ is a subset of $[k]^{<\mathbb{N}}$ of the form \[ \{w\}\cup \big\{w^{\smallfrown}w_0(a_0)^{\smallfrown}...^{\smallfrown}w_{n}(a_n): n\in \{0,...,m-1\} \text{ and } a_0,...,a_n\in [k]\big\} \] where $w$ is a word over $k$ and $(w_n)_{n=0}^{m-1}$ is a finite sequence of left variable words over $k$. We study the behavior of a family of measurable events in a probability space indexed by the elements of a Carlson-Simpson tree of sufficiently large dimension. Specifically we show the following. For every integer $k\geq 2$, every $0<\varepsilon\leq 1$ and every integer $n\geq 1$ there exists a strictly positive constant $\theta(k,\varepsilon,n)$ with the following property. If $m$ is a given positive integer, then there exists an integer $\mathrm{Cor}(k,\varepsilon,m)$ such that for every Carlson--Simpson tree $T$ of $[k]^{<\mathbb{N}}$ of dimension at least $\mathrm{Cor}(k,\varepsilon,m)$ and every family $\{A_t:t\in T\}$ of measurable events in a probability space $(\Omega,\Sigma,\mu)$ satisfying $\mu(A_t)\geq \varepsilon$ for every $t\in T$, there exists a Carlson--Simpson tree $S$ of dimension $m$ with $S\subseteq T$ and such that for every nonempty $F\subseteq S$ we have \[\mu\Big(\bigcap_{t\in F} A_t\Big) \geq \theta(k,\varepsilon,|F|). \] The proof is based, among others, on the density version of the Carlson--Simpson Theorem established recently by the authors, as well as, on a partition result -- of independent interest -- closely related to the work of T. J. Carlson, and H. Furstenberg and Y. Katznelson. The argument is effective and yields explicit lower bounds for the constants $\theta(k,\varepsilon,n)$.