Two Erdős problems on lacunary sequences: chromatic number and Diophantine approximation

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• Yuval Peres (Microsoft Research)
• Thursday 03 November 2011, 14:30-15:30
• MR12.

If you have a question about this talk, please contact Andrew Thomason.

Abstract: Let {n_k} be a lacunary sequence, i.e., the ratio of successive elements of the sequence is at least some q>1. In 1987, Erdős asked for the chromatic number of a graph G on the integers, where two integers are connected by an edge iff their difference is in the sequence {n_k}. Y.Katznelson found a connection via a to a Diophantine approximation problem: finding irrationals x such that n_k times x is at least r>0 away from the integers for all k. In joint work with W.Schlag, we improve Katznelson’s bounds for both problems using the Lovasz local lemma. It is still an unsolved problem to obtain matching upper and lower bounds.

This talk is part of the Combinatorics Seminar series.

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