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Pseudorandomness of a Random Kronecker Sequence

  • Eda Cesaratto
  • Brigitte Vallée
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7256)

Abstract

We study two randomness measures for the celebrated Kronecker sequence \({\cal S}(\alpha)\) formed by the fractional parts of the multiples of a real α. The first measure is the well-known discrepancy, whereas the other one, the Arnold measure, is less popular. Both describe the behaviour of the truncated sequence \({\cal S}_T(\alpha)\) formed with the first T terms, for T → ∞. We perform a probabilistic study of the pseudorandomness of the sequence \({\cal S}(\alpha)\) (discrepancy and Arnold measure), and we give estimates of their mean values in two probabilistic settings : the input α may be either a random real or a random rational. The results exhibit strong similarities between the real and rational cases; they also show the influence of the number T of truncated terms, via its relation to the continued fraction expansion of α.

Keywords

Probabilistic Setting Randomness Measure Real Case Transfer Operator Dirichlet Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Eda Cesaratto
    • 1
  • Brigitte Vallée
    • 2
  1. 1.CONICET and Univ. Nac. de Gral. SarmientoBuenos AiresArgentina
  2. 2.Laboratoire GREYCCNRS UMR 6072 and Université de CaenCaenFrance

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