On the Compatibility of Binary Sequences

An ordered pair of semi‐infinite binary sequences (η,ξ) is said to be compatible if there is a way of removing a certain number (possibly infinite) of ones from η and zeroes from ξ that would map both sequences to the same semi‐infinite sequence. This notion was introduced by Peter Winkler, who also posed the following question: η and ξ being independent i.i.d. Bernoulli sequences with parameters p′ and p, respectively, does there 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′ very close to . In the positive direction, we construct, for any ? > 0, a deterministic binary sequence η_? whose set of zeroes has Hausdorff dimension larger than 1 ? ? and such that ?_p {ξ : (η_?,ξ) is compatible } > 0 for p small enough, where ?_p stands for the product Bernoulli measure with parameter p. © 2014 Wiley Periodicals, Inc.