On series Σc k f(kx) and Khinchin’s conjecture

István Berkes; Michel Weber

doi:10.1007/s11856-014-0036-0

On series Σc k f(kx) and Khinchin’s conjecture

István Berkes, Michel Weber

Institute of Statistics (5060)

Research output: Contribution to journal › Article › peer-review

Abstract

We prove the optimality of a criterion of Koksma (1953) in Khinchin’s conjecture on strong uniform distribution. This verifies a claim of Bourgain (1988) and leads also to a near optimal a.e. convergence condition for series Σ ∞ k=1 c k f(kx) with f ∈ L 2. Finally, we show that under mild regularity conditions on the Fourier coefficients of f, the Khinchin conjecture is valid assuming only f ∈ L 2.

Original language	English
Pages (from-to)	593-609
Journal	Israel Journal of Mathematics
Volume	201
Issue number	2
DOIs	https://doi.org/10.1007/s11856-014-0036-0
Publication status	Published - Feb 2014

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Information, Communication & Computing

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Basic - Fundamental (Grundlagenforschung)

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10.1007/s11856-014-0036-0

on_series_and_khinchins_conjectureAccepted author manuscript, 143 KBLicence: CC BY 4.0

Cite this

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title = "On series Σc k f(kx) and Khinchin{\textquoteright}s conjecture",

abstract = "We prove the optimality of a criterion of Koksma (1953) in Khinchin{\textquoteright}s conjecture on strong uniform distribution. This verifies a claim of Bourgain (1988) and leads also to a near optimal a.e. convergence condition for series Σ ∞ k=1 c k f(kx) with f ∈ L 2. Finally, we show that under mild regularity conditions on the Fourier coefficients of f, the Khinchin conjecture is valid assuming only f ∈ L 2.",

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AU - Weber, Michel

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N2 - We prove the optimality of a criterion of Koksma (1953) in Khinchin’s conjecture on strong uniform distribution. This verifies a claim of Bourgain (1988) and leads also to a near optimal a.e. convergence condition for series Σ ∞ k=1 c k f(kx) with f ∈ L 2. Finally, we show that under mild regularity conditions on the Fourier coefficients of f, the Khinchin conjecture is valid assuming only f ∈ L 2.

AB - We prove the optimality of a criterion of Koksma (1953) in Khinchin’s conjecture on strong uniform distribution. This verifies a claim of Bourgain (1988) and leads also to a near optimal a.e. convergence condition for series Σ ∞ k=1 c k f(kx) with f ∈ L 2. Finally, we show that under mild regularity conditions on the Fourier coefficients of f, the Khinchin conjecture is valid assuming only f ∈ L 2.

UR - http://www.ma.huji.ac.il/~ijmath/index_pdf/201.pdf

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DO - 10.1007/s11856-014-0036-0

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EP - 609

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 2

ER -

On series Σc k f(kx) and Khinchin’s conjecture

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Analytic Combinatorics: Analytic Combinatorics and Probabilistic Number Theory

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On series Σc k f(kx) and Khinchin’s conjecture

Abstract

Fields of Expertise

Treatment code (Nähere Zuordnung)

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Projects

Analytic Combinatorics: Analytic Combinatorics and Probabilistic Number Theory

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