Distribution Functions for Subsequences of Generalized Van Der Corput Sequences

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Abstract

For an integer b > 1 let (φb(n))n≥0 denote the van der Corput sequence base in b in [0, 1). Answering a question of O. Strauch, C. Aistleitner and M. Hofer showed that the distribution function of (φb(n), φb(n + 1), . . . , φb(n + s − 1))n≥0 on [0, 1)s exists and is a copula. The first and third authors of the present paper showed that this phenomenon extends to a broad class of subsequences of the van der Corput sequence. In this result we extend this paper still further and show that this phenomenon is also true for more general numeration systems based on the beta expansion of W. Parry and A. Rényi.

[AH] AISTLEITNER, C.-HOFER, M.: On the limit distribution of consequtive elements of the van der Corput sequence, Unif. Distrib. Theory 8 (2013), no. 1, 89-96.

[AN] ASMAR, N. H.-NAIR, R.: Certain averages on the a-adic numbers, Proc. Amer. Math. Soc. 114 (1992) no. 1, 21-28.

[B] BRAUER, A.: On algebraic equations with all but one root in the interior of the unit circle, Math. Nachr. 4, (1951) 250-257.

[CFS] CORNFELD, I. P.-FORMIN, S. V.-SINAI, YA. G.: Ergodic Theory, Springer-Verlag, Berlin, 1982.

[DP] DICK, J.-PILLICHSHAMMER, F.: Digital nets and Sequences, Discrepancy and Quasi-Monte Carlo Integration, Cambridge University Press, Cambridge, 2010.

[DT] DRMOTA, M.-TICHY, R. F.: Sequences, Discrepancies and Applications. In: Lecture Notes in Mathematics Vol. 1651, Springer-Verlag, Berlin, 1997.

[FMS] FIALOVÁ, J.-MIŠ´IK, L.-STRAUCH, O.: An asymptotic distribution function of the three-dimensional shifted van der Corput sequence, Appl. Math. 5 (2014), 2334-2359. Published Online August 2014 in SciRes: http://www.scirp.org/journal/amhttp://dx.doi.org/10.4236/am.2014.515227

[FS] FIALOVÁ, J.-STRAUCH, O.: On two-dimensional sequences composed by one-dimensional uniformly distibuted sequences, Unif. Distrib. Theory 6 (2011), no. 1, 101-125.

[GHL] GRABNE, P. J.-HELLEKALEK, P.-LIARDET, P.: The dynamical point of view of low discrepancy sequences, Unif. Distrib. Theory, 7 (2012), no. 1, 11-70.

[HJLN] HANČL, J.-JAŠŠOVÁ, A.-LERTCHOOSAKUL, P.-NAIR, R.: On the metric theory of p-adic continued fractions, Indag. Math. (N.S.) 24 (2013), no. 1, 42-56.

[HN] HELLEKALEK, P.-NIEDERREITER, H.: Constructions of uniformly distributed sequences using the b-adic method, Unif. Distrib. Theory 6 (2011), no. 1, 185-200.

[HR] HEWITT, E.-ROSS, K. A.: Abstract Harmonic Analysis (2nd edition), Grundlehren der Mathematishchen Wissenschaftenenm. Vol. 115. A Series of Comprehensive Studies in Mathematics. Springer Verlag, Berlin, 1979.

[HIT] HOFER, M.-IACÒ, M. R.-TICHY, R.: Ergodic properties of β-adic Halton sequences Ergodic Theory Dyn. Syst. 35 (2015), no. 3, 895-909.

[JLN] JAŠŠOVÁ, A.-LERTCHOOSAKUL, P.-NAIR, R.: On variants of the Halton Sequence, Monat. Math. 180 (2016), no. 4, 743-764.

[HKLP] HOFER, R.-KRITZER, P.-LARCHER, G.-PILLICHSHAMMER, F.: Distribution properties of generalized van der Corput-Halton sequences and their subsequences, Int. J. Number Theory 5 (2009), no. 4, 719-746.

[KN] KUIPERS, L.-NIEDERREITER, H.: Uniform Distribution of Sequences, John Wiley & Sons, New York 1974.

[LN] LERTCHOOSAKUL, P.-NAIR, R.: Distribution functions for subsequences of the van der Corput sequence, Indag.Math. (N.S.) 24 (2013), no. 3, 593-601.

[O] OXTOBY, J. C.: Ergodic sets. Bull. Amer. Math. Soc. 58 (1952), 116-136.

[N1] NAIR, R.: On asymptotic distribution of the a-adic integers, Proc. Indian Acad. Sci. Math. Sci. 107 (1997), no. 4, 363-376.

[Ni] NINOMIYA, S.: Construction a new class of low-discrepancy sequences by using the β-adic transformation., Math. Comput. Simulation 47 (1998), no. (2), 403-418.9

[P] PARRY, W.: On β-expansions of real numbers, Acta. Math. Acad. Sci. Hungar. 11 (1960), 401-416.

Uniform distribution theory

The Journal of Slovak Academy of Sciences

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