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The sixth International Conference on Uniform Distribution Theory (UDT 2018) CIRM, Luminy, Marseilles, France, October 1–5, 2018

Volume 13 (2018): Issue 2 (December 2018)

Volume 13 (2018): Issue 1 (June 2018)

Volume 12 (2017): Issue 2 (December 2017)

Volume 12 (2017): Issue 1 (June 2017)

Volume 11 (2016): Issue 2 (December 2016)

Volume 11 (2016): Issue 1 (June 2016)

Journal Details
Format
Journal
eISSN
2309-5377
First Published
30 Dec 2013
Publication timeframe
2 times per year
Languages
English

Search

Volume 13 (2018): Issue 1 (June 2018)

Journal Details
Format
Journal
eISSN
2309-5377
First Published
30 Dec 2013
Publication timeframe
2 times per year
Languages
English

Search

7 Articles
access type Open Access

Questions Around the Thue-Morse Sequence

Published Online: 20 Jul 2018
Page range: 1 - 25

Abstract

Abstract

We intend to unroll the surprizing properties of the Thue-Morse sequence with a harmonic analysis point of view, and mention in passing some related open questions.

Keywords

  • Thue-Morse
  • Rudin-Shapiro
  • sum of digits
  • trigonometric polynomial
  • correlation measure
  • singular measure
access type Open Access

Motzkin’s Maximal Density and Related Chromatic Numbers

Published Online: 20 Jul 2018
Page range: 27 - 45

Abstract

Abstract

This paper concerns the problem of determining or estimating the maximal upper density of the sets of nonnegative integers S whose elements do not differ by an element of a given set M of positive integers. We find some exact values and some bounds for the maximal density when the elements of M are generalized Fibonacci numbers of odd order. The generalized Fibonacci sequence of order r is a generalization of the well known Fibonacci sequence, where instead of starting with two predetermined terms, we start with r predetermined terms and each term afterwards is the sum of r preceding terms. We also derive some new properties of the generalized Fibonacci sequence of order r. Furthermore, we discuss some related coloring parameters of distance graphs generated by the set M.

Keywords

  • maximal density
  • generalized Fibonacci numbers
  • fractional coloring
  • circular coloring

MSC 2010

  • 11B05
  • 11B39
access type Open Access

On the Discrepancy of Two Families of Permuted Van der Corput Sequences

Published Online: 20 Jul 2018
Page range: 47 - 64

Abstract

Abstract

A permuted van der Corput sequence Sbσ$S_b^\sigma$ in base b is a one-dimensional, infinite sequence of real numbers in the interval [0, 1), generation of which involves a permutation σ of the set {0, 1,..., b − 1}. These sequences are known to have low discrepancy DN, i.e. t(Sbσ):=limsupNDN(Sbσ)/logN$t\left({S_b^\sigma } \right): = {\rm{lim}}\,{\rm{sup}}_{N \to \infty } D_N \left({S_b^\sigma } \right)/{\rm{log}}\,N$ is finite. Restricting to prime bases p we present two families of generating permutations. We describe their elements as polynomials over finite fields 𝔽p in an explicit way. We use this characterization to obtain bounds for t(Spσ)$t\left({S_p^\sigma } \right)$ for permutations σ in these families. We determine the best permutations in our first family and show that all permutations of the second family improve the distribution behavior of classical van der Corput sequences in the sense that t(Spσ)<t(Spid)$t\left({S_p^\sigma } \right) < t\left({S_p^{id} } \right)$.

Keywords

  • van der Corput sequence
  • extreme discrepancy
  • permutation
  • permutation polynomial
  • Carlitz rank

MSC 2010

  • 11K38
  • 11K06
  • 11T06
access type Open Access

Metrical Star Discrepancy Bounds for Lacunary Subsequences of Digital Kronecker-Sequences and Polynomial Tractability

Published Online: 20 Jul 2018
Page range: 65 - 86

Abstract

Abstract

The star discrepancy DN*(𝒫)$D_N^* \left( {\cal P} \right)$ is a quantitative measure for the irregularity of distribution of a finite point set 𝒫 in the multi-dimensional unit cube which is intimately related to the integration error of quasi-Monte Carlo algorithms. It is known that for every integer N ≥ 2 there are point sets 𝒫 in [0, 1)d with |𝒫| = N and DN*(𝒫)=O((logN)d-1/N)$D_N^* \left( {\cal P} \right) = O\left( {\left( {\log \,N} \right)^{d - 1} /N} \right)$. However, for small N compared to the dimension d this asymptotically excellent bound is useless (e.g., for N ≤ ed−1).

In 2001 it has been shown by Heinrich, Novak, Wasilkowski and Woźniakowski that for every integer N ≥ 2there exist point sets 𝒫 in [0, 1)d with |𝒫| = N and DN*(𝒫)Cd/N$D_N^* \left( {\cal P} \right) \le C\sqrt {d/N}$. Although not optimal in an asymptotic sense in N, this upper bound has a much better (and even optimal) dependence on the dimension d.

Unfortunately the result by Heinrich et al. and also later variants thereof by other authors are pure existence results and until now no explicit construction of point sets with the above properties is known. Quite recently Löbbe studied lacunary subsequences of Kronecker’s (nα)-sequence and showed a metrical discrepancy bound of the form Cd(logd)/N$C\sqrt {d\left({\log \,d} \right)/N}$ with implied absolute constant C> 0 independent of N and d.

In this paper we show a corresponding result for digital Kronecker sequences, which are a non-archimedean analog of classical Kronecker sequences.

Keywords

  • star discrepancy
  • digital Kronecker-sequence
  • polynomial tractability
  • quasi-Monte Carlo

MSC 2010

  • 11K38
  • 11K31
  • 11K45
access type Open Access

An Extension of the Digital Method Based on b-Adic Integers

Published Online: 20 Jul 2018
Page range: 87 - 107

Abstract

Abstract

We introduce a hybridization of digital sequences with uniformly distributed sequences in the domain of b-adic integers, ℤb,b ∈ℕ \ {1}, by using such sequences as input for generating matrices. The generating matrices are then naturally required to have finite row-lengths. We exhibit some relations of the ‘classical’ digital method to our extended version, and also give several examples of new constructions with their respective quality assessments in terms of t, T and discrepancy.

Keywords

  • quasi-Monte Carlo methods
  • construction
  • digital method
  • digit expansion
  • -adic integers

MSC 2010

  • 11J71
  • 11K16
  • 11K38
  • 11F85
access type Open Access

Construction of Uniformly Distributed Linear Recurring Sequences Modulo Powers of 2

Published Online: 20 Jul 2018
Page range: 109 - 129

Abstract

Abstract

The aim of the present paper is to provide the background to construct linear recurring sequences with uniform distribution modulo 2s. The theory is developed and an algorithm based on the achieved results is given. The constructed sequences may have arbitrary large period length depending only on the computational power of the used machines.

Keywords

  • uniform distribution
  • construction
  • long period
  • large numbers

MSC 2010

  • 11B37
  • 11B50
access type Open Access

Weak Universality Theorem on the Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function from a Beatty Sequence

Published Online: 20 Jul 2018
Page range: 131 - 146

Abstract

Abstract

In this paper, we prove a discrete analogue of Voronin’s early finite-dimensional approximation result with respect to terms from a given Beatty sequence and make use of Taylor approximation in order to derive a weak universality statement.

Keywords

  • Universality
  • Riemann zeta-function
  • Beatty sequences

MSC 2010

  • 11M99
  • 30K10
7 Articles
access type Open Access

Questions Around the Thue-Morse Sequence

Published Online: 20 Jul 2018
Page range: 1 - 25

Abstract

Abstract

We intend to unroll the surprizing properties of the Thue-Morse sequence with a harmonic analysis point of view, and mention in passing some related open questions.

Keywords

  • Thue-Morse
  • Rudin-Shapiro
  • sum of digits
  • trigonometric polynomial
  • correlation measure
  • singular measure
access type Open Access

Motzkin’s Maximal Density and Related Chromatic Numbers

Published Online: 20 Jul 2018
Page range: 27 - 45

Abstract

Abstract

This paper concerns the problem of determining or estimating the maximal upper density of the sets of nonnegative integers S whose elements do not differ by an element of a given set M of positive integers. We find some exact values and some bounds for the maximal density when the elements of M are generalized Fibonacci numbers of odd order. The generalized Fibonacci sequence of order r is a generalization of the well known Fibonacci sequence, where instead of starting with two predetermined terms, we start with r predetermined terms and each term afterwards is the sum of r preceding terms. We also derive some new properties of the generalized Fibonacci sequence of order r. Furthermore, we discuss some related coloring parameters of distance graphs generated by the set M.

Keywords

  • maximal density
  • generalized Fibonacci numbers
  • fractional coloring
  • circular coloring

MSC 2010

  • 11B05
  • 11B39
access type Open Access

On the Discrepancy of Two Families of Permuted Van der Corput Sequences

Published Online: 20 Jul 2018
Page range: 47 - 64

Abstract

Abstract

A permuted van der Corput sequence Sbσ$S_b^\sigma$ in base b is a one-dimensional, infinite sequence of real numbers in the interval [0, 1), generation of which involves a permutation σ of the set {0, 1,..., b − 1}. These sequences are known to have low discrepancy DN, i.e. t(Sbσ):=limsupNDN(Sbσ)/logN$t\left({S_b^\sigma } \right): = {\rm{lim}}\,{\rm{sup}}_{N \to \infty } D_N \left({S_b^\sigma } \right)/{\rm{log}}\,N$ is finite. Restricting to prime bases p we present two families of generating permutations. We describe their elements as polynomials over finite fields 𝔽p in an explicit way. We use this characterization to obtain bounds for t(Spσ)$t\left({S_p^\sigma } \right)$ for permutations σ in these families. We determine the best permutations in our first family and show that all permutations of the second family improve the distribution behavior of classical van der Corput sequences in the sense that t(Spσ)<t(Spid)$t\left({S_p^\sigma } \right) < t\left({S_p^{id} } \right)$.

Keywords

  • van der Corput sequence
  • extreme discrepancy
  • permutation
  • permutation polynomial
  • Carlitz rank

MSC 2010

  • 11K38
  • 11K06
  • 11T06
access type Open Access

Metrical Star Discrepancy Bounds for Lacunary Subsequences of Digital Kronecker-Sequences and Polynomial Tractability

Published Online: 20 Jul 2018
Page range: 65 - 86

Abstract

Abstract

The star discrepancy DN*(𝒫)$D_N^* \left( {\cal P} \right)$ is a quantitative measure for the irregularity of distribution of a finite point set 𝒫 in the multi-dimensional unit cube which is intimately related to the integration error of quasi-Monte Carlo algorithms. It is known that for every integer N ≥ 2 there are point sets 𝒫 in [0, 1)d with |𝒫| = N and DN*(𝒫)=O((logN)d-1/N)$D_N^* \left( {\cal P} \right) = O\left( {\left( {\log \,N} \right)^{d - 1} /N} \right)$. However, for small N compared to the dimension d this asymptotically excellent bound is useless (e.g., for N ≤ ed−1).

In 2001 it has been shown by Heinrich, Novak, Wasilkowski and Woźniakowski that for every integer N ≥ 2there exist point sets 𝒫 in [0, 1)d with |𝒫| = N and DN*(𝒫)Cd/N$D_N^* \left( {\cal P} \right) \le C\sqrt {d/N}$. Although not optimal in an asymptotic sense in N, this upper bound has a much better (and even optimal) dependence on the dimension d.

Unfortunately the result by Heinrich et al. and also later variants thereof by other authors are pure existence results and until now no explicit construction of point sets with the above properties is known. Quite recently Löbbe studied lacunary subsequences of Kronecker’s (nα)-sequence and showed a metrical discrepancy bound of the form Cd(logd)/N$C\sqrt {d\left({\log \,d} \right)/N}$ with implied absolute constant C> 0 independent of N and d.

In this paper we show a corresponding result for digital Kronecker sequences, which are a non-archimedean analog of classical Kronecker sequences.

Keywords

  • star discrepancy
  • digital Kronecker-sequence
  • polynomial tractability
  • quasi-Monte Carlo

MSC 2010

  • 11K38
  • 11K31
  • 11K45
access type Open Access

An Extension of the Digital Method Based on b-Adic Integers

Published Online: 20 Jul 2018
Page range: 87 - 107

Abstract

Abstract

We introduce a hybridization of digital sequences with uniformly distributed sequences in the domain of b-adic integers, ℤb,b ∈ℕ \ {1}, by using such sequences as input for generating matrices. The generating matrices are then naturally required to have finite row-lengths. We exhibit some relations of the ‘classical’ digital method to our extended version, and also give several examples of new constructions with their respective quality assessments in terms of t, T and discrepancy.

Keywords

  • quasi-Monte Carlo methods
  • construction
  • digital method
  • digit expansion
  • -adic integers

MSC 2010

  • 11J71
  • 11K16
  • 11K38
  • 11F85
access type Open Access

Construction of Uniformly Distributed Linear Recurring Sequences Modulo Powers of 2

Published Online: 20 Jul 2018
Page range: 109 - 129

Abstract

Abstract

The aim of the present paper is to provide the background to construct linear recurring sequences with uniform distribution modulo 2s. The theory is developed and an algorithm based on the achieved results is given. The constructed sequences may have arbitrary large period length depending only on the computational power of the used machines.

Keywords

  • uniform distribution
  • construction
  • long period
  • large numbers

MSC 2010

  • 11B37
  • 11B50
access type Open Access

Weak Universality Theorem on the Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function from a Beatty Sequence

Published Online: 20 Jul 2018
Page range: 131 - 146

Abstract

Abstract

In this paper, we prove a discrete analogue of Voronin’s early finite-dimensional approximation result with respect to terms from a given Beatty sequence and make use of Taylor approximation in order to derive a weak universality statement.

Keywords

  • Universality
  • Riemann zeta-function
  • Beatty sequences

MSC 2010

  • 11M99
  • 30K10

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