On the compatibility of binary sequences

TL;DR

This paper investigates the compatibility of semi-infinite binary sequences, showing that for certain deterministic sequences with large zero sets, there is a positive probability of compatibility with random Bernoulli sequences, extending previous results.

Contribution

It constructs deterministic sequences with large zero sets that are compatible with random Bernoulli sequences with small parameter, answering a question posed by Peter Winkler.

Findings

Compatible pairs exist with positive probability for certain deterministic sequences.

Sequences with large Hausdorff dimension of zeroes can be compatible with Bernoulli sequences.

Compatibility depends on the parameters of the Bernoulli sequences and the structure of the deterministic sequence.

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 $p$.