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

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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.

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Authors and Affiliations

  1. Institute of Statistics, Graz University of Technology, Kopernikusgasse 24/III, 8010, Graz, Austria

    István Berkes

  2. IRMA, 10 rue du Général Zimmer, 67084, Strasbourg Cedex, France

    Michel Weber

Authors
  1. István Berkes
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  2. Michel Weber
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Correspondence to István Berkes.

Additional information

Research supported by FWF Grant P 24302-N18 and OTKA Grant K 81928.

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Berkes, I., Weber, M. On series Σc k f(kx) and Khinchin’s conjecture. Isr. J. Math. 201, 593–609 (2014). https://doi.org/10.1007/s11856-014-0036-0

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  • DOI: https://doi.org/10.1007/s11856-014-0036-0

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