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Published in:
2012 | OriginalPaper | Chapter
Pseudorandomness of a Random Kronecker Sequence
Authors : Eda Cesaratto, Brigitte Vallée
Published in: LATIN 2012: Theoretical Informatics
Publisher: Springer Berlin Heidelberg
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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
.