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Pseudorandomness of a Random Kronecker Sequence

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Part of the Lecture Notes in Computer Science book series (LNTCS,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

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Cesaratto, E., Vallée, B. (2012). Pseudorandomness of a Random Kronecker Sequence. In: Fernández-Baca, D. (eds) LATIN 2012: Theoretical Informatics. LATIN 2012. Lecture Notes in Computer Science, vol 7256. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29344-3_14

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  • DOI: https://doi.org/10.1007/978-3-642-29344-3_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-29343-6

  • Online ISBN: 978-3-642-29344-3

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