On the compatibility of binary sequences

Abstract

An ordered pair of semi-infinite binary sequences $(η,ξ)$ is said to be compatible if there is a way of removing a certain number (possibly infinite) of ones from $η$ and zeroes from $ξ$, whichwould map both sequences to the same semi-infinite sequence. This notion was introduced by Peter Winkler, who also posed the following question: $η$ and $ξ$ being independent i.i.d. Bernoulli sequences with parameters $p^\prime$ and $p$ respectively, does it exist $(p', p)$ so that the set of compatible pairs has positive measure? It is known that this does not happen for $p$ and $p^\prime$ very close to 1/2. In the positive direction, we construct, for any $ε> 0$, a deterministic binary sequence $η_ε$ whose set of zeroes has Hausdorff dimension larger than $1-ε$, and such that $\mathbb{P}_p {ξ\colon (η_ε,ξ) \text {is compatible}} > 0$ for $p$ small enough, where $\mathbb{P}_p$ stands for the product Bernoulli measure with parameter $p$.