Clairvoyant embedding in one dimension
Probability
2014-03-24 v3 Combinatorics
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.
Cite
@article{arxiv.1204.4897,
title = {Clairvoyant embedding in one dimension},
author = {Peter Gacs},
journal= {arXiv preprint arXiv:1204.4897},
year = {2014}
}
Comments
49 pages. Some errors corrected. arXiv admin note: substantial text overlap with arXiv:math/0109152