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On hybrid sequences built from Niederreiter-Halton sequences and Kronecker sequences. (English) Zbl 1234.11096

In this article, the authors discuss the distribution properties of hybrid sequences whose components stem from Niederreiter-Halton sequences and Kronecker sequences. Let \((x_n)_{n\geq 0}\) be an \(s\)-dimensional Niederreiter-Halton sequence and \((y_n)_{n\geq 0}\) be a \(d\)-dimensional Kronecker sequence. They prove that the \((s+d)\)-dimensional sequence \((z_n)_{n\geq 0}\) with \(z_n=(x_n, y_n)\) is uniformly distributed modulo one if and only if \((x_n)_{n\geq 0}\) and \((y_n)_{n\geq 0}\) are both uniformly distributed modulo one. Furthermore the authors derive a quantitative result on the discrepancy of the hybrid sequences.

MSC:

11K06 General theory of distribution modulo \(1\)
11K38 Irregularities of distribution, discrepancy
11L07 Estimates on exponential sums
Full Text: DOI

References:

[1] Pillichshammer, Unif. Distrib. Theory 2 pp 1– (2007)
[2] Spanier, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing pp 121– (1995) · doi:10.1007/978-1-4612-2552-2_6
[3] DOI: 10.4064/aa138-4-8 · Zbl 1268.11102 · doi:10.4064/aa138-4-8
[4] Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods (1992) · Zbl 0761.65002 · doi:10.1137/1.9781611970081
[5] DOI: 10.1007/BF01294651 · Zbl 0626.10045 · doi:10.1007/BF01294651
[6] DOI: 10.4064/aa141-4-5 · Zbl 1219.11112 · doi:10.4064/aa141-4-5
[7] Niederreiter, Applications of Number Theory to Numerical Analysis pp 203– (1972) · doi:10.1016/B978-0-12-775950-0.50011-X
[8] Lemieux, Monte Carlo and Quasi-Monte Carlo Sampling (2009)
[9] DOI: 10.1142/S1793042109002328 · Zbl 1188.11038 · doi:10.1142/S1793042109002328
[10] DOI: 10.2307/2154922 · Zbl 0829.11039 · doi:10.2307/2154922
[11] DOI: 10.1016/j.jnt.2008.05.012 · Zbl 1219.11111 · doi:10.1016/j.jnt.2008.05.012
[12] Kuipers, Uniform Distribution of Sequences (1974)
[13] DOI: 10.1007/978-3-642-04107-5_27 · Zbl 1228.11123 · doi:10.1007/978-3-642-04107-5_27
[14] DOI: 10.4064/aa138-2-6 · Zbl 1228.11114 · doi:10.4064/aa138-2-6
[15] Hofer, Unif. Distrib. Theory 2 pp 35– (2007)
[16] Drmota, Sequences, Discrepancies and Applications (1997) · Zbl 0877.11043 · doi:10.1007/BFb0093404
[17] DOI: 10.1006/jnth.2000.2628 · Zbl 0990.11053 · doi:10.1006/jnth.2000.2628
[18] Dick, Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration (2010) · Zbl 1282.65012 · doi:10.1017/CBO9780511761188
[19] DOI: 10.1007/s00605-009-0150-y · Zbl 1273.11117 · doi:10.1007/s00605-009-0150-y
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