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Nilsequences and multiple correlations along subsequences

Part of: Ergodic theory

Published online by Cambridge University Press:  08 October 2018

ANH NGOC LE
ANH NGOC LE*
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA email anhle@math.northwestern.edu
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Abstract

The results of Bergelson, Host and Kra, and Leibman state that a multiple polynomial correlation sequence can be decomposed into a sum of a nilsequence (a sequence defined by evaluating a continuous function along an orbit in a nilsystem) and a null sequence (a sequence that goes to zero in density). We refine their results by proving that the null sequence goes to zero in density along polynomials evaluated at primes and along the Hardy sequence $(\lfloor n^{c}\rfloor )$. In contrast, given a rigid sequence, we construct an example of a correlation whose null sequence does not go to zero in density along that rigid sequence. As a corollary of a lemma in the proof, the formula for the pointwise ergodic average along polynomials of primes in a nilsystem is also obtained.

Keywords

classical ergodic theorynumber theorymultiple correlationnilsequencenull sequence

MSC classification

Primary: 37A45: Relations with number theory and harmonic analysis
Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 6 , June 2020 , pp. 1634 - 1654
Copyright
© Cambridge University Press, 2018

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