ABSTRACT

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