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On the law of the iterated logarithm for the discrepancy of 〈n k x〉

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Abstract.

By a well known result of Philipp (1975), the discrepancy D N (ω) of the sequence (n k ω) k≥1 mod 1 satisfies the law of the iterated logarithm under the Hadamard gap condition n k + 1/n k ≥ q > 1 (k = 1, 2, …). Recently Berkes, Philipp and Tichy (2006) showed that this result remains valid, under Diophantine conditions on (n k ), for subexpenentially growing (n k ), but in general the behavior of (n k ω) becomes very complicated in the subexponential case. Using a different norming factor depending on the density properties of the sequence (n k ), in this paper we prove a law of the iterated logarithm for the discrepancy D N (ω) for subexponentially growing (n k ) without number theoretic assumptions.

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Authors and Affiliations

  1. Graz University of Technology, Graz, Austria

    Christoph Aistleitner & Istvan Berkes

Authors
  1. Christoph Aistleitner
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  2. Istvan Berkes
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Additional information

C. Aistleitner, Research supported by FWF grant S9603-N13.

I. Berkes, Research supported by FWF grant S9603-N13 and OTKA grants K 61052 and K 67961.

Authors’ addresses: C. Aistleitner, Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria; I. Berkes, Institute of Statistics, Graz University of Technology, Steyrergasse 17/IV, 8010 Graz, Austria

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Aistleitner, C., Berkes, I. On the law of the iterated logarithm for the discrepancy of 〈n k x〉. Monatsh Math 156, 103–121 (2009). https://doi.org/10.1007/s00605-008-0549-x

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  • DOI: https://doi.org/10.1007/s00605-008-0549-x

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