TY - JOUR

T1 - Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

AU - BERGELSON, V.

AU - KUŁAGA-PRZYMUS, J.

AU - LEMAŃCZYK, M.

AU - Richter, Florian Karl

N1 - Funding Information:
Proof of Theorem 5.5. The following argument is analogous to the one used in the proof of Theorem 5.1. Let x ∈ AN be WRAP and let f ∈ C(Xx) and z ∈ Xx be arbitrary. It follows from Proposition 3.6 that z|N is WRAP. Therefore, equation (5.8) follows from (5.9). □ Acknowledgements. We thank the anonymous referee for many helpful comments. The first author gratefully acknowledges the support of the NSF under grant DMS-1500575. The second author’s research is supported by Narodowe Centrum Nauki UMO-2014/15/B/ST1/03736 and by the Foundation for Polish Science (FNP). The third author’s research was supported by Narodowe Centrum Nauki UMO-2014/15/B/ST1/03736 and the EU grant ‘AOS’, FP7-PEOPLE-2012-IRSES, No. 318910.

PY - 2019/9/1

Y1 - 2019/9/1

N2 - A set is called rational if it is well approximable by finite unions of arithmetic progressions, meaning that for every 0 > there exists a set , where , such that for all ; (b) is an averaging set of polynomial single recurrence; (c) is an averaging set of polynomial multiple recurrence. As an application, we show that if is rational and divisible, then for any set , where and Examples of rational sets include many classical sets of number-theoretical origin such as the set of squarefree numbers, the set of abundant numbers, or sets of the form , then the following are equivalent: (a) is divisible, i.e. 0$]] is Euler's totient function. We investigate the combinatorial and dynamical properties of rational sets and obtain new results in ergodic Ramsey theory. Among other things, we show that if is a rational set with 0$]] with 0]]> and any polynomials , , which satisfy and for all , there exists 0 > such that the set STIX eqnarray > has positive lower density. Ramsey-theoretical applications naturally lead to problems in symbolic dynamics, which involve rationally almost periodic sequences (sequences whose level-sets are rational). We prove that if is a finite alphabet, is rationally almost periodic, denotes the left-shift on and then is a generic point for an -invariant probability measure on such that the measure-preserving system is ergodic and has rational discrete spectrum.

AB - A set is called rational if it is well approximable by finite unions of arithmetic progressions, meaning that for every 0 > there exists a set , where , such that for all ; (b) is an averaging set of polynomial single recurrence; (c) is an averaging set of polynomial multiple recurrence. As an application, we show that if is rational and divisible, then for any set , where and Examples of rational sets include many classical sets of number-theoretical origin such as the set of squarefree numbers, the set of abundant numbers, or sets of the form , then the following are equivalent: (a) is divisible, i.e. 0$]] is Euler's totient function. We investigate the combinatorial and dynamical properties of rational sets and obtain new results in ergodic Ramsey theory. Among other things, we show that if is a rational set with 0$]] with 0]]> and any polynomials , , which satisfy and for all , there exists 0 > such that the set STIX eqnarray > has positive lower density. Ramsey-theoretical applications naturally lead to problems in symbolic dynamics, which involve rationally almost periodic sequences (sequences whose level-sets are rational). We prove that if is a finite alphabet, is rationally almost periodic, denotes the left-shift on and then is a generic point for an -invariant probability measure on such that the measure-preserving system is ergodic and has rational discrete spectrum.

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U2 - 10.1017/etds.2017.130

DO - 10.1017/etds.2017.130

M3 - Article

AN - SCOPUS:85041576597

VL - 39

SP - 2332

EP - 2383

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 9

ER -