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

Moreira, Joel; Richter, Florian K.

doi:10.1017/etds.2017.61

Cited by 12 publications

(17 citation statements)

References 38 publications

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“…For example, Bergelson, Host and Kra [6] analyze this component to prove a generalization of Khintchine's theorem. Moreira and Richter [29] show that this component arises from a system whose spectrum is contained in the spectrum of the original system.…”

Section: Introductionmentioning

confidence: 99%

Nilsequences and multiple correlations along subsequences

Le¹

2018

Ergod. Th. Dynam. Sys.

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The results of Bergelson-Host-Kra and Leibman say 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 Hardy sequence (⌊n c ⌋). On the other hand, given a rigid sequence, we construct an example of correlation whose null sequence does not approach 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.

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Section: Introductionmentioning

confidence: 99%

Nilsequences and multiple correlations along subsequences

Le¹

2018

Ergod. Th. Dynam. Sys.

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“…If in (5) one replaces the optimal lower bound µ(A) k+1 − with 0, then the set is syndetic for any k ∈ N. This was proved in [19] for k = 2 and k = 3, and for larger k, this is essentially the content of [8, Corollary 6.5]; see also [26]. Theorems 1.2, 1.3 and 1.4 will be proved in Section 3.…”

Section: 2mentioning

confidence: 92%

“…In order to show that A = B = 0, we first show that B is a positive scalar multiple of A. Once we do that, we have from (27) that B = v 1 v 4 v 2 v 3 A and hence, equality between the first and last quantities from (26)…”

mentioning

confidence: 99%

Optimal lower bounds for multiple recurrence

Donoso¹,

Le²,

Moreira³

et al. 2019

Ergod. Th. Dynam. Sys.

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Let (X, B, µ, T ) be an ergodic measure preserving system, A ∈ B and > 0. We study the largeness of sets of the formsatisfies q(1) or q(−1) = 0 and pn denotes the n-th prime; or when f is a certain Hardy field sequence. If T q is ergodic for some q ∈ N, then for all r ∈ Z, S is syndetic if f (n) = qn + r.For fi(n) = ain, where ai are distinct integers, we show that S can be empty for k ≥ 4, and for k = 3 we found an interesting relation between the largeness of S and the abundance of solutions to certain linear equations in sparse sets of integers. We also provide some partial results when the fi are distinct polynomials. arXiv:1809.06912v2 [math.DS]

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“…where π(N) denotes the number of primes less than or equal to N. See [7,14,35] for more information.…”

Section: Density Problemsmentioning

confidence: 99%

Topological characteristic factors and nilsystems

Glasner¹,

Huang²,

Shao³

et al. 2020

Preprint

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We prove that the maximal infinite step pro-nilfactor X ∞ of a minimal dynamical system (X, T ) is the topological characteristic factor in a certain sense. Namely, we show that by an almost one to one modification of π : X → X ∞ , the induced open extension π * : X * → X * ∞ has the following property: for x in a dense G δ set of X * , the orbit closureUsing results derived from the above fact, we are able to answer several open questions: (1) if (X, T k ) is minimal for some k ≥ 2, then for any d ∈ N and any 0 ≤ j < k there is a sequence(3) for any d ∈ N and any minimal system, which is an open extension of its maximal distal factor, RP [d] = AP [d] , where the latter is the regionally proximal relation of order d along arithmetic progressions.

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

Cited by 12 publications

References 38 publications

Nilsequences and multiple correlations along subsequences

Nilsequences and multiple correlations along subsequences

Optimal lower bounds for multiple recurrence

Topological characteristic factors and nilsystems

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