Discrepancy of LS-sequences of partitions and points

Carbone, Ingrid

doi:10.1007/s10231-011-0208-z

Cited by 27 publications

(56 citation statements)

References 6 publications

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“…If k = 2, then we obtain the definition of the classical LS-sequence of partitions introduced by C a r b o n e [4]. Given a sequence of partitions we can assign a sequence of points by ordering the left endpoints of the intervals in the partition.…”

Section: ò ø óò 3 (Generalized Ls-sequence Of Partitions)º Letmentioning

confidence: 99%

“…One issue of this paper is to find an ordering that yields good upper bounds for the discrepancy of the sequence. In the case that k = 2 the corresponding LS-sequences of points, denoted by (ξ n L,S ) n∈N , have been introduced by C a r b o n e [4] who proved that whenever L ≥ S there exists a positive constant k 1 such that…”

Section: ò ø óò 3 (Generalized Ls-sequence Of Partitions)º Letmentioning

confidence: 99%

“…This problem still open in [4] can be solved by the numeration approach presented by A i s t l e i t n e r et.al. [2].…”

Section: ò ø óò 3 (Generalized Ls-sequence Of Partitions)º Letmentioning

confidence: 99%

“…[2]. However the main focus of the present paper is to generalize the construction of classical LS-sequences of points by C a r b o n e [4] in the following sense: we introduce a numeration system for the integers which is the generalization of the numeration system introduced by A i s t l e i t n e r et.al. [2] and, by a radical inverse construction, similar as the construction of the van der Corput sequence, we study the obtained sequence of points.…”

Section: ò ø óò 3 (Generalized Ls-sequence Of Partitions)º Letmentioning

confidence: 99%

“…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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confidence: 99%

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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: ò ø óò 3 (Generalized Ls-sequence Of Partitions)º Letmentioning

confidence: 99%

Section: ò ø óò 3 (Generalized Ls-sequence Of Partitions)º Letmentioning

confidence: 99%

“…This problem still open in [4] can be solved by the numeration approach presented by A i s t l e i t n e r et.al. [2].…”

Section: ò ø óò 3 (Generalized Ls-sequence Of Partitions)º Letmentioning

confidence: 99%

Section: ò ø óò 3 (Generalized Ls-sequence Of Partitions)º Letmentioning

confidence: 99%

mentioning

confidence: 99%

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Discrepancy of Generalized LS-Sequences

Iacò

Ziegler

2017

Uniform Distribution Theory

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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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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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Discrepancy of LS-sequences of partitions and points

Abstract: In this paper we give a precise estimate of the discrepancy of a class of uniformly distributed sequences of partitions. Among them we found a large class having low discrepancy (which means of order 1 N ). One of them is the Kakutani-Fibonacci sequence.

Cited by 27 publications

References 6 publications

Discrepancy of Generalized LS-Sequences

Discrepancy of Generalized LS-Sequences

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

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

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