Pseudorandomness of a Random Kronecker Sequence

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


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.


Probabilistic Setting Randomness Measure Real Case Transfer Operator Dirichlet Series 
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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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