Ergodic properties of -adic Halton sequences

Hofer, Markus; Iacò, Maria Rita; Tichy, Robert F.

doi:10.1017/etds.2013.70

Cited by 16 publications

(37 citation statements)

References 28 publications

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“…This sequence, also called Kakutani-Fibonacci sequence, has been also analysed in detail in [5,9] in the frame of ergodic theory where it has been shown that it can be obtained as the orbit of an ergodic transformation.…”

Section: Discrepancy Bounds For Classical Ls-sequencesmentioning

confidence: 99%

Discrepancy of Generalized LS-Sequences

Iacò

Ziegler

2017

Uniform Distribution Theory

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ABSTRACT. The LS-sequences are a parametric family of sequences of points in the unit interval. They were introduced by Carbone [4], who also proved that under an appropriate choice of the parameters L and S, such sequences are lowdiscrepancy. The aim of the present paper is to provide explicit constants in the bounds of the discrepancy of LS-sequences. Further, we generalize the construction of Carbone [4] and construct a new class of sequences of points in the unit interval, the generalized LS-sequences.

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Section: Discrepancy Bounds For Classical Ls-sequencesmentioning

confidence: 99%

Discrepancy of Generalized LS-Sequences

Iacò

Ziegler

2017

Uniform Distribution Theory

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“…In particular, when L = S = 1, the Kakutani-Fibonacci (1, 1)-sequence coincides with the β-adic van der Corput sequence where β = Φ is the golden ratio, i.e. the positive root of x 2 − x − 1 (see also [18] for more details). β-adic sequences and L S-sequences provide, in dimension 1, low-discrepancy sequences.…”

Section: Introductionmentioning

confidence: 89%

“…They have been studied quite extensively. For good references on the subject we suggest [3,18,22,23]. For the original definition of van der Corput sequence see [25].…”

Section: Introductionmentioning

confidence: 99%

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Comparison Between LS-Sequences and $$\beta $$ β -Adic van der Corput Sequences

Carbone

2016

Springer Proceedings in Mathematics &Amp; Statistics

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In 2011 the author introduced a generalization of van der Corput sequences, the so called L S-sequences defined for integers L , S such that L ≥ 1, S ≥ 0, L + S ≥ 2, and γ ∈ ]0, 1[ is the positive solution of Sγ 2 + Lγ = 1. These sequences coincide with the classical van der Corput sequences whenever S = 0, are uniformly distributed for all L , S and have low discrepancy when L ≥ S. In this paper we compare the L S-sequences and the β-adic van der Corput sequences where β > 1 is the Pisot root of x 2 − Lx − L. Using a suitable numeration system G = {G n } n≥0 , where the base sequence is the linear recurrence of order two, G n+2 = LG n+1 + LG n , with initial conditions G 0 = 1 and G 1 = L + 1, we prove that when L = S the (L , L)-sequence with Lγ 2 + Lγ = 1 and the β-adic van der Corput sequence with β = 1/γ and β 2 = Lβ + L can be obtained from each other by a permutation. In particular for β = Φ, the golden ratio, the β-adic van der Corput sequence coincides with the Kakutani-Fibonacci sequence obtained for L = S = 1, which has been already studied.

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“…It is worth noting here that the van der Corput sequence and some one-dimensional low-discrepancy sequences with respect to the Cantor expansion were studied in [2] and [5]. Note also that it was mentioned in [8] about the Halton sequence in a more generalized numeration system than the Cantor expansion, called the G-expansion; however, the paper aimed to study the Halton sequence in some fixed non-integer bases and did not touch on the Halton sequence with respect to dynamical bases.…”

Section: Introductionmentioning

confidence: 99%