Multiple recurrence along sparse sequences over thick sets

Multiple recurrence along sparse sequences over thick sets

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Joel Moreira, Northwestern University
Fine Hall 1001

Khintchine’s recurrence theorem states that the set of optimal return times in a measure preserving dynamical system is syndetic, i.e., has bounded gaps. In 1977, as part of his ergodic-theoretic proof of Szemeredi’s theorem on arithmetic progressions, Furstenberg established a partial extension of Khintchine’s result by showing that the set of multiple return times is also syndetic. Multiple recurrence has since been established along many different types of sequences, including polynomial sequences and sequences derived from functions in a Hardy field. However, they don’t always lead to syndetic return time sets. In my talk I will describe joint work with Vitaly Bergelson and Florian Richter where we establish that for a general class of non-polynomial sparse sequences, the set of return times still possesses interesting combinatorial properties, and in particular it satisfies a weak form of syndeticity and is thick, i.e. contains arbitrarily long intervals of integers. Via Furstenberg’s correspondence principle our work leads to novel variants of Szemeredi’s theorem.