Clairvoyant embedding in one dimension

TL;DR

This paper proves that for two independent coin-tossing sequences, there exists a positive probability that one sequence can be embedded into the other with bounded gaps, answering an open question in the field.

Contribution

The paper demonstrates the existence of a positive probability that one random binary sequence can be embedded into another with bounded gaps, extending previous hierarchical methods.

Findings

Existence of an m such that Y is m-embeddable into X with positive probability

Generalization of hierarchical method for dependent percolation

Answers an open question in sequence embedding theory

Abstract

Let v, w be infinite 0-1 sequences, and m a positive integer. We say that w is m-embeddable in v, if there exists an increasing sequence n_{i} of integers with n_{0}=0, such that 0< n_{i} - n_{i-1} < m, w(i) = v(n_i) for all i > 0. Let X and Y be independent coin-tossing sequences. We will show that there is an m with the property that Y is m-embeddable into X with positive probability. This answers a question that was open for a while. The proof generalizes somewhat the hierarchical method of an earlier paper of the author on dependent percolation.