Ergodic averaging sequences

Boshernitzan, Michael; Kolesnik, Grigori; Quas, Anthony; Wierdl, Μáté

doi:10.1007/bf02791497

Cited by 57 publications

(90 citation statements)

References 14 publications

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“…A similar phenomenon was observed (with regards to mean convergence) in the paper [9] alluded to above.…”

Section: The Following Are Functions From Fsupporting

confidence: 84%

“…Hardy fields also appear in [9] where the authors study the generalizations of the classical von Neumann and Birkhoff ergodic theorems. While there is practically no overlap between our paper and [9], the two papers demonstrate, each in its own way, that Hardy fields provide a natural framework for extensions of some of the familiar ergodic results.…”

Section: The Following Are Functions From Fmentioning

confidence: 99%

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Weak mixing implies weak mixing of higher orders along tempered functions

Bergelson¹,

Inger²

2009

Ergod. Th. Dynam. Sys.

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“…A similar phenomenon was observed (with regards to mean convergence) in the paper [9] alluded to above.…”

Section: The Following Are Functions From Fsupporting

confidence: 84%

Section: The Following Are Functions From Fmentioning

confidence: 99%

Weak mixing implies weak mixing of higher orders along tempered functions

Bergelson¹,

Inger²

2009

Ergod. Th. Dynam. Sys.

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“…[7]) we get that the limit (22) is bounded from below by a positive constant c 2 that depends only on h dm X = f dμ (and is independent of r 0 ). Combining this with (17) and (18), we get that…”

Section: Conclusion Of the Argumentmentioning

confidence: 89%

“…2 Proof of Theorem 1.2. If |a(x) − cp(x)| log x for every p ∈ Z[x]and c ∈ R, then by[17] the sequence ([a(n)]) n∈N is good for the ergodic theorem,11 and the result follows.Therefore, we can assume that a(x) = cp(x) + b(x) for some p ∈ Z[x], c ∈ R, and b ∈ H that satisfies 1 ≺ b(x) ≺ x. Furthermore,we can assume that b(x) 0 for large x (the other case can be treated similarly). If c = 0, then the range of the sequence ([b(n)]) n∈N contains all large enough integers (see Lemma 5.2) and so forms a set of recurrence.…”

mentioning

confidence: 81%

A Hardy field extension of Szemerédi's theorem

Frantzikinakis

Wierdl

2009

Advances in Mathematics

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In 1975 Szemerédi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common difference of the arithmetic progression can be a square, a cube, or more generally of the form p(n) where p(n) is any integer polynomial with zero constant term. We produce a variety of new results of this type related to sequences that are not polynomial. We show that the common difference of the progression in Szemerédi's theorem can be of the form [n δ ] where δ is any positive real number and [x] denotes the integer part of x. More generally, the common difference can be of the form [a(n)] where a(x) is any function that is a member of a Hardy field and satisfies a(x)/x k → ∞ and a(x)/x k+1 → 0 for some non-negative integer k. The proof combines a new structural result for Hardy sequences, techniques from ergodic theory, and some recent equidistribution results of sequences on nilmanifolds. Published by Elsevier Inc. MSC: primary 37A45; secondary 28D05, 05D10, 11B25

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“…, Δ 0 M k satisfy conditions (b) and (c) in §2. 3. Therefore, if s is sufficiently large, there exist positive integers y 1 , .…”

Section: Proof Of Theorem a When R(x)mentioning

confidence: 93%