On the compatibility of binary sequences

Abstract

An ordered pair of semi-infinite binary sequences $(\eta,\xi)$ is said to be compatible if there is a way of removing a certain number (possibly infinite) of ones from $\eta$ and zeroes from $\xi$, whichwould map both sequences to the same semi-infinite sequence. This notion was introduced by Peter Winkler, who also posed the following question: $\eta$ and $\xi$ 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 $\epsilon > 0$, a deterministic binary sequence $\eta_\epsilon$ whose set of zeroes has Hausdorff dimension larger than $1-\epsilon$, and such that $\mathbb{P}_p {\xi\colon (\eta_\epsilon,\xi) \text {is compatible}} > 0$ for $p$ small enough, where $\mathbb{P}_p$ stands for the product Bernoulli measure with parameter $p$.