Dynamical system generated by algebraic method and low discrepancy sequences

Mori, M.; Mori, Masaki

doi:10.1515/mcma-2012-0012

Cited by 3 publications

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“…Discrepancy bounds for a higher dimensional extension of [23] are given in [22]. Note that the construction in [22] is different from that in the present article.…”

Section: Definitionmentioning

confidence: 85%

Ergodic properties of -adic Halton sequences

Hofer¹,

Iacò²,

Tichy³

2013

Ergod. Th. Dynam. Sys.

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We investigate a parametric extension of the classical s-dimensional Halton sequence, where the bases are special Pisot numbers. In a onedimensional setting the properties of such sequences have already been investigated by several authors [5,8,23,28]. We use methods from ergodic theory to in order to investigate the distribution behavior of multidimensional versions of such sequences. As a consequence it is shown that the Kakutani-Fibonacci transformation is uniquely ergodic.

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“…Discrepancy bounds for a higher dimensional extension of [23] are given in [22]. Note that the construction in [22] is different from that in the present article.…”

Section: Definitionmentioning

confidence: 85%

Ergodic properties of -adic Halton sequences

Hofer¹,

Iacò²,

Tichy³

2013

Ergod. Th. Dynam. Sys.

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“…Another recent construction of low-discrepancy sequences using ergodic theory was given by father and son Mori [15]. This work extended to arbitrary dimensions the earlier constructions in [13] and [14] for the dimensions s = 2 and s = 3, respectively.…”

Section: And 33])mentioning

confidence: 86%

“…This work extended to arbitrary dimensions the earlier constructions in [13] and [14] for the dimensions s = 2 and s = 3, respectively. The paper [15] is hard to read, first of all because of unconventional terminology: "dimension of polynomial" instead of "degree of polynomial", "normal matrix" instead of "nonsingular matrix", and the expression "area of words" is unclear. There is also notation that is inconvenient for the reader, e.g.…”

Section: And 33])mentioning

confidence: 99%

“…an overuse of the letter A: the letter A stands for a finite alphabet, A is also the name of a rule (see [15, p. 332]),Â is the finite field with 2 d elements,Ã appears on p. 331 of [15] in a crucial context as an undefined symbol (probably a misprint-but this misprint makes it impossible to say on which set the fundamental transformationF is defined), the A i denote column vectors, and A 1 denotes the set of bits. There is a similar excessive use of the letters w and W. As in [12], the discrepancy bound in [15] is without explicit constants. It would be desirable to improve the presentation in the paper [15] in order to make this interesting work more accessible.…”

Section: And 33])mentioning

confidence: 99%

“…There is a similar excessive use of the letters w and W. As in [12], the discrepancy bound in [15] is without explicit constants. It would be desirable to improve the presentation in the paper [15] in order to make this interesting work more accessible.…”

Section: And 33])mentioning

confidence: 99%

See 2 more Smart Citations

Recent constructions of low-discrepancy sequences

Niederreiter

2017

Mathematics and Computers in Simulation

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Discrepancy Bounds for β $$\boldsymbol{\beta }$$ -adic Halton Sequences

Thuswaldner

2017

Number Theory – Diophantine Problems, Uniform Distribution and Applications

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Van der Corput and Halton sequences are well-known low-discrepancy sequences. Almost twenty years ago Ninomiya defined analogues of van der Corput sequences for βnumeration and proved that they also form low-discrepancy sequences if β is a Pisot number. Only very recently Robert Tichy and his co-authors succeeded in proving that β-adic Halton sequences are equidistributed for certain parameters β = (β 1 , . . . , βs) using methods from ergodic theory. In the present paper we continue this research and give discrepancy estimates for β-adic Halton sequences for which the components β i are m-bonacci numbers. Our methods are quite different and use dynamical and geometric properties of Rauzy fractals that allow to relate β-adic Halton sequences to rotations on high dimensional tori. The discrepancies of these rotations can then be estimated by classical methods relying on W. M. Schmidt's Subspace Theorem.∞ j=0 ε j (n)G j ,

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Dynamical system generated by algebraic method and low discrepancy sequences

Cited by 3 publications

References 0 publications

Ergodic properties of -adic Halton sequences

Ergodic properties of -adic Halton sequences

Recent constructions of low-discrepancy sequences

Discrepancy Bounds for β $$\boldsymbol{\beta }$$ -adic Halton Sequences

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