The structure of strongly stationary systems

Frantzikinakis, Nikos

doi:10.1007/bf02789313

Cited by 23 publications

(29 citation statements)

References 23 publications

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“…It can then be shown that the resulting system ( X, µ, T ) has additional structure, namely it is strongly stationary (see Definition 5.1). The structure of strongly stationary systems was completely determined in [43] and [22], where it was shown that they satisfy properties (i) and (ii) of Theorem 1.6. Unfortunately, we do not know how to establish total ergodicity of the ergodic components of Furstenberg systems of the Liouville function (for the Möbius function this property is not even true).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The logarithmic Sarnak conjecture for ergodic weights

2018

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The Möbius disjointness conjecture of Sarnak states that the Möbius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero topological entropy. We verify the logarithmically averaged variant of this conjecture for a large class of systems, which includes all uniquely ergodic systems with zero entropy. One consequence of our results is that the Liouville function has super-linear block growth. Our proof uses a disjointness argument and the key ingredient is a structural result for measure preserving systems naturally associated with the Möbius and the Liouville function. We prove that such systems have no irrational spectrum and their building blocks are infinite-step nilsystems and Bernoulli systems. To establish this structural result we make a connection with a problem of purely ergodic nature via some identities recently obtained by Tao. In addition to an ergodic structural result of Host and Kra, our analysis is guided by the notion of strong stationarity which was introduced by Furstenberg and Katznelson in the early 90's and naturally plays a central role in the structural analysis of measure preserving systems associated with multiplicative functions.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

The logarithmic Sarnak conjecture for ergodic weights

2018

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“…In the following we will use I to denote the subgroup of T generated by a subset I ⊂ T. Subsets of R are tacitly identified with their projections mod 1 onto T. Theorem 2.4 (cf. [8,Theorem 6.4]). Let q, r ∈ N, and let (X, B, µ, T ) be an ergodic measure preserving system whose discrete spectrum…”

Section: Remark 22mentioning

confidence: 99%

A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems

Moreira¹,

Richter

2017

Ergod. Th. Dynam. Sys.

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We investigate how spectral properties of a measure preserving system (X, B, µ, T ) are reflected in the multiple ergodic averages arising from that system. For certain sequences a : N → N we provide natural conditions on the spectrum σ(T ) such that for all f 1 , .In particular, our results apply to infinite arithmetic progressions a(n) = qn+r, Beatty sequences a(n) = ⌊θn+γ⌋, the sequence of squarefree numbers a(n) = q n , and the sequence of prime numbers a(n) = p n .We also obtain a new refinement of Szemerédi's theorem via Furstenberg's correspondence principle.

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“…For convergence, the proof also carries over, using the general result on convergence of linear averages in [15]. Furthermore, the generalization of Theorem 5 holds, using the description of the characteristic factors in [15] and the higher analog of identity (11) in [6].…”

Section: Further Generalizationsmentioning

confidence: 98%

Multiple recurrence and convergence for sequences related to the prime numbers

Frantzikinakis

Kra

2007

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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106

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Abstract. For any measure preserving system (X, X , µ, T ) and A ∈ X with µ(A) > 0, we show that there exist infinitely many primes p such that µ`A ∩ T −(p−1) A ∩ T −2(p−1) A´> 0 (the same holds with p − 1 replaced by p + 1). Furthermore, we show the existence of the limit in L 2 (µ) of the associated ergodic average over the primes. A key ingredient is a recent result of Green and Tao on the von Mangoldt function. A combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form p − 1 (or p + 1) for some prime p.

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The structure of strongly stationary systems

Cited by 23 publications

References 23 publications

The logarithmic Sarnak conjecture for ergodic weights

The logarithmic Sarnak conjecture for ergodic weights

A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems

Multiple recurrence and convergence for sequences related to the prime numbers

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