Continued fraction algorithms, functional operators, and structure constants
Theoretical Computer Science
An Average-Case Analysis of the Gaussian Algorithm for Lattice Reduction
Combinatorics, Probability and Computing
Regularity of the Euclid Algorithm; application to the analysis of fast GCD Algorithms
Journal of Symbolic Computation
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We study two randomness measures for the celebrated Kronecker sequence S(α) 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 ST(α) formed with the first T terms, for T→∞. We perform a probabilistic study of the pseudorandomness of the sequence S(α) (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.