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Volume 17 (2022): Issue 1 (December 2022)

Volume 16 (2021): Issue 2 (December 2021)

Volume 16 (2021): Issue 1 (June 2021)

Volume 15 (2020): Issue 2 (December 2020)

Volume 15 (2020): Issue 1 (June 2020)

Volume 14 (2019): Issue 2 (December 2019)

Volume 14 (2019): Issue 1 (June 2019)
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 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

11 Articles
access type Open Access

Pierre Liardet (1943–2014)

Published Online: 13 Jan 2017
Page range: i - xi

Abstract

access type Open Access

An Improved Bound for the Star Discrepancy of Sequences in the Unit Interval

Published Online: 13 Jan 2017
Page range: 1 - 14

Abstract

Abstract

It is known that there is a constant c > 0 such that for every sequence x1, x2, . . . in [0, 1) we have for the star discrepancy DN*$D_N^* $ of the first N elements of the sequence that NDN*clogN$ND_N^* \ge c \cdot \log N$ holds for infinitely many N. Let c be the supremum of all such c with this property. We show c > 0.065664679 . . . , thereby slightly improving the estimates known until now.

Keywords

  • uniform distribution
  • discrepancy

MSC 2010

  • 11K38
  • 11K06
access type Open Access

Uniform Distribution of the Sequence of Balancing Numbers Modulo m

Published Online: 13 Jan 2017
Page range: 15 - 21

Abstract

Abstract

The balancing numbers and the balancers were introduced by Behera et al. in the year 1999, which were obtained from a simple diophantine equation. The goal of this paper is to investigate the moduli for which all the residues appear with equal frequency with a single period in the sequence of balancing numbers. Also, it is claimed that, the balancing numbers are uniformly distributed modulo 2, and this holds for all other powers of 2 as well. Further, it is shown that the balancing numbers are not uniformly distributed over odd primes.

Keywords

  • Balancing numbers
  • Balancers
  • Uniform distribution
  • Periodicity of balancing numbers

MSC 2010

  • 11B37
  • 11B39
access type Open Access

Distribution of Leading Digits of Numbers

Published Online: 13 Jan 2017
Page range: 23 - 45

Abstract

Abstract

Applying the theory of distribution functions of sequences we find the relative densities of the first digits also for sequences xn not satisfying Benford’s law. Especially for sequence xn = nr, n = 1, 2, . . . and xn=pnr$x_n = p_n^r $, n = 1, 2, . . ., where pn is the increasing sequence of all primes and r > 0 is an arbitrary real. We also add rate of convergence to such densities.

Keywords

  • Benford’s law
  • distribution function
  • prime number

MSC 2010

  • 11K06
  • 11K31
access type Open Access

On the Pseudorandomness of the Liouville Function of Polynomials over a Finite Field

Published Online: 13 Jan 2017
Page range: 47 - 58

Abstract

Abstract

We study several pseudorandom properties of the Liouville function and the Möbius function of polynomials over a finite field. More precisely, we obtain bounds on their balancedness as well as their well-distribution measure, correlation measure, and linear complexity profile.

Keywords

  • polynomials
  • finite fields
  • irreducible factors
  • pseudorandom sequence
  • balancedness
  • well-distribution
  • correlation measure
  • linear complexity
  • polynomial Liouville function
  • polynomial Möbius function

MSC 2010

  • 11K45
  • 11T06
  • 11T24
  • 11T71
access type Open Access

On Strong Normality

Published Online: 13 Jan 2017
Page range: 59 - 78

Abstract

Abstract

We introduce the concept of strong normality by defining strong normal numbers and provide various properties of these numbers, including the fact that almost all real numbers are strongly normal.

Keywords

  • normal numbers
  • uniform distribution modulo 1

MSC 2010

  • 11K16
  • 11N37
access type Open Access

On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions

Published Online: 13 Jan 2017
Page range: 79 - 139

Abstract

Abstract

Let n ≥ 2 be an integer and denote by θn the real root in (0, 1) of the trinomial Gn(X) = −1 + X + Xn. The sequence of Perron numbers (θn1)n2$(\theta _n^{ - 1} )_{n \ge 2} $ tends to 1. We prove that the Conjecture of Lehmer is true for {θn1|n2}$\{ \theta _n^{ - 1} |n \ge 2\} $ by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the roots θn, zj,n, of Gn(X) lying in |z| < 1, as a function of n, j only. This method, not yet applied to Lehmer’s problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measures M(Gn)=M(θn)=M(θn1)${\rm{M}}(G_n ) = {\rm{M}}(\theta _n ) = {\rm{M}}(\theta _n^{ - 1} )$ of the trinomials Gn as a function of n only, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for {θn1|n2}$\{ \theta _n^{ - 1} |n \ge 2\} $, with a minoration of the house , and a minoration of the Mahler measure M(Gn) better than Dobrowolski’s one. The angular regularity of the roots of Gn, near the unit circle, and limit equidistribution of the conjugates, for n tending to infinity (in the sense of Bilu, Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context of the Erdős-Turán-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.

Keywords

  • Mahler measure
  • trinomial
  • Lehmer Conjecture
  • asymptotic expansion
  • divergent series
  • Perron number
  • Pisot number
  • Schinzel-Zassenhaus conjecture
  • Smyth conjecture
  • Lind-Boyd Conjecture
  • Boyd Conjecture
  • Erdős-Turán-Mignotte-Amoroso
  • discrepancy function
  • limit equidistribution

MSC 2010

  • 11C08
  • 11G50
  • 11K16
  • 11K26
  • 11K38
  • 11Q05
  • 11R06
  • 11R09
  • 30B10
  • 30C15
access type Open Access

On Hausdorff Dimensions Related to Sets with Given Asymptotic and Gap Densities

Published Online: 13 Jan 2017
Page range: 141 - 157

Abstract

Abstract

For a set A of positive integers a1< a2< · · ·, let d(A), d¯(A)$\overline d (A)$ denote its lower and upper asymptotic densities. The gap density is defined as λ(A)=limsupnan+1an$\lambda (A) = \lim \;{\rm sup} _{n \to \infty } {{a_{n + 1} } \over {a_n }}$. The paper investigates the class 𝒢(α, β, γ) of all sets A with d(A) = α, d¯(A)=β$\overline d (A) = \beta $ and λ(A) = γ for given α, β, γ with 0 ≤ α ≤ β ≤ 1 ≤ γ and αγ ≤ β. Using the classical dyadic mapping ϱ(A)=n=1χA(n)2n$\varrho (A) = \sum\nolimits_{n = 1}^\infty {{{\chi _A (n)} \over {2^n }}} $, where χA is the characteristic function of A, the main result of the paper states that the ϱ-image set ϱ𝒢(α, β, γ) has the Hausdorff dimension dimϱ𝒢(α,β,γ)=min{δ(α),δ(β),1γmaxσ[αγ,β]δ(σ)},$$\dim \varrho \cal {G}(\alpha ,\beta ,\gamma ) = \min \left\{ {\delta (\alpha ),\delta (\beta ), { 1 \over \gamma }\mathop {\max }\limits_{\sigma \in [\alpha \gamma ,\beta ]} \delta (\sigma )} \right\},$$ where δ is the entropy function δ(x)=xlog2x(1x)log2(1x).$$\delta (x) = - x\log _2 x - (1 - x)\;\log _2 (1 - x).$$

Keywords

  • Sequences of integers
  • lower asymptotic density
  • upper asymptotic density
  • gap density
  • Hausdorff dimension

MSC 2010

  • 11B05
  • 11K55
access type Open Access

Pair Correlations and Random Walks on the Integers

Published Online: 13 Jan 2017
Page range: 159 - 164

Abstract

Abstract

The paper gives conditions for a sequence of fractional parts of real numbers ({anx})n=1$\left( {\{ a_n x\} } \right)_{n = 1}^\infty $ to satisfy a pair correlation estimate. Here x is a fixed nonzero real number and (an)n=1$\left( {a_n } \right)_{n = 1}^\infty $ is a random walk on the integers.

Keywords

  • pair correlation
  • random walks

MSC 2010

  • 11K38
  • 60G50
access type Open Access

On Small Sets of Distribution Functions of Ratio Block Sequences

Published Online: 13 Jan 2017
Page range: 165 - 174

Abstract

Abstract

There are various methods how to describe and characterize distribution of elements of sets of positive integers. One of the most interesting is that using the set of all distribution functions of the corresponding ratio block sequence introduced in [Strauch, O.—Tóth, J.T.: Publ. Math. Debrecen 58 (2001), no. 4, 751–778]. In the present paper we give some sufficient conditions under which this set is small in a metric sense. As a corollary we obtain a new characterization of the case of asymptotic distribution.

Keywords

  • asymptotic distribution function
  • uniform distribution
  • block sequence

MSC 2010

  • 11B05
access type Open Access

Linear Recursive Odometers and Beta-Expansions

Published Online: 13 Jan 2017
Page range: 175 - 186

Abstract

Abstract

The aim of this paper is to study the connection between different properties related to β-expansions. In particular, the relation between two conditions, both ensuring purely discrete spectrum of the odometer, is analyzed. The first one is the so-called Hypothesis B for the G-odometers and the second one is denoted by (QM) and it has been introduced in the framework of tilings associated to Pisot β-numerations.

Keywords

  • beta-expansions
  • odometer
  • purely discrete spectrum
  • finiteness property

MSC 2010

  • 11A63
  • 11K16
  • 11B37
  • 28D05
11 Articles
access type Open Access

Pierre Liardet (1943–2014)

Published Online: 13 Jan 2017
Page range: i - xi

Abstract

access type Open Access

An Improved Bound for the Star Discrepancy of Sequences in the Unit Interval

Published Online: 13 Jan 2017
Page range: 1 - 14

Abstract

Abstract

It is known that there is a constant c > 0 such that for every sequence x1, x2, . . . in [0, 1) we have for the star discrepancy DN*$D_N^* $ of the first N elements of the sequence that NDN*clogN$ND_N^* \ge c \cdot \log N$ holds for infinitely many N. Let c be the supremum of all such c with this property. We show c > 0.065664679 . . . , thereby slightly improving the estimates known until now.

Keywords

  • uniform distribution
  • discrepancy

MSC 2010

  • 11K38
  • 11K06
access type Open Access

Uniform Distribution of the Sequence of Balancing Numbers Modulo m

Published Online: 13 Jan 2017
Page range: 15 - 21

Abstract

Abstract

The balancing numbers and the balancers were introduced by Behera et al. in the year 1999, which were obtained from a simple diophantine equation. The goal of this paper is to investigate the moduli for which all the residues appear with equal frequency with a single period in the sequence of balancing numbers. Also, it is claimed that, the balancing numbers are uniformly distributed modulo 2, and this holds for all other powers of 2 as well. Further, it is shown that the balancing numbers are not uniformly distributed over odd primes.

Keywords

  • Balancing numbers
  • Balancers
  • Uniform distribution
  • Periodicity of balancing numbers

MSC 2010

  • 11B37
  • 11B39
access type Open Access

Distribution of Leading Digits of Numbers

Published Online: 13 Jan 2017
Page range: 23 - 45

Abstract

Abstract

Applying the theory of distribution functions of sequences we find the relative densities of the first digits also for sequences xn not satisfying Benford’s law. Especially for sequence xn = nr, n = 1, 2, . . . and xn=pnr$x_n = p_n^r $, n = 1, 2, . . ., where pn is the increasing sequence of all primes and r > 0 is an arbitrary real. We also add rate of convergence to such densities.

Keywords

  • Benford’s law
  • distribution function
  • prime number

MSC 2010

  • 11K06
  • 11K31
access type Open Access

On the Pseudorandomness of the Liouville Function of Polynomials over a Finite Field

Published Online: 13 Jan 2017
Page range: 47 - 58

Abstract

Abstract

We study several pseudorandom properties of the Liouville function and the Möbius function of polynomials over a finite field. More precisely, we obtain bounds on their balancedness as well as their well-distribution measure, correlation measure, and linear complexity profile.

Keywords

  • polynomials
  • finite fields
  • irreducible factors
  • pseudorandom sequence
  • balancedness
  • well-distribution
  • correlation measure
  • linear complexity
  • polynomial Liouville function
  • polynomial Möbius function

MSC 2010

  • 11K45
  • 11T06
  • 11T24
  • 11T71
access type Open Access

On Strong Normality

Published Online: 13 Jan 2017
Page range: 59 - 78

Abstract

Abstract

We introduce the concept of strong normality by defining strong normal numbers and provide various properties of these numbers, including the fact that almost all real numbers are strongly normal.

Keywords

  • normal numbers
  • uniform distribution modulo 1

MSC 2010

  • 11K16
  • 11N37
access type Open Access

On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions

Published Online: 13 Jan 2017
Page range: 79 - 139

Abstract

Abstract

Let n ≥ 2 be an integer and denote by θn the real root in (0, 1) of the trinomial Gn(X) = −1 + X + Xn. The sequence of Perron numbers (θn1)n2$(\theta _n^{ - 1} )_{n \ge 2} $ tends to 1. We prove that the Conjecture of Lehmer is true for {θn1|n2}$\{ \theta _n^{ - 1} |n \ge 2\} $ by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the roots θn, zj,n, of Gn(X) lying in |z| < 1, as a function of n, j only. This method, not yet applied to Lehmer’s problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measures M(Gn)=M(θn)=M(θn1)${\rm{M}}(G_n ) = {\rm{M}}(\theta _n ) = {\rm{M}}(\theta _n^{ - 1} )$ of the trinomials Gn as a function of n only, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for {θn1|n2}$\{ \theta _n^{ - 1} |n \ge 2\} $, with a minoration of the house , and a minoration of the Mahler measure M(Gn) better than Dobrowolski’s one. The angular regularity of the roots of Gn, near the unit circle, and limit equidistribution of the conjugates, for n tending to infinity (in the sense of Bilu, Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context of the Erdős-Turán-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.

Keywords

  • Mahler measure
  • trinomial
  • Lehmer Conjecture
  • asymptotic expansion
  • divergent series
  • Perron number
  • Pisot number
  • Schinzel-Zassenhaus conjecture
  • Smyth conjecture
  • Lind-Boyd Conjecture
  • Boyd Conjecture
  • Erdős-Turán-Mignotte-Amoroso
  • discrepancy function
  • limit equidistribution

MSC 2010

  • 11C08
  • 11G50
  • 11K16
  • 11K26
  • 11K38
  • 11Q05
  • 11R06
  • 11R09
  • 30B10
  • 30C15
access type Open Access

On Hausdorff Dimensions Related to Sets with Given Asymptotic and Gap Densities

Published Online: 13 Jan 2017
Page range: 141 - 157

Abstract

Abstract

For a set A of positive integers a1< a2< · · ·, let d(A), d¯(A)$\overline d (A)$ denote its lower and upper asymptotic densities. The gap density is defined as λ(A)=limsupnan+1an$\lambda (A) = \lim \;{\rm sup} _{n \to \infty } {{a_{n + 1} } \over {a_n }}$. The paper investigates the class 𝒢(α, β, γ) of all sets A with d(A) = α, d¯(A)=β$\overline d (A) = \beta $ and λ(A) = γ for given α, β, γ with 0 ≤ α ≤ β ≤ 1 ≤ γ and αγ ≤ β. Using the classical dyadic mapping ϱ(A)=n=1χA(n)2n$\varrho (A) = \sum\nolimits_{n = 1}^\infty {{{\chi _A (n)} \over {2^n }}} $, where χA is the characteristic function of A, the main result of the paper states that the ϱ-image set ϱ𝒢(α, β, γ) has the Hausdorff dimension dimϱ𝒢(α,β,γ)=min{δ(α),δ(β),1γmaxσ[αγ,β]δ(σ)},$$\dim \varrho \cal {G}(\alpha ,\beta ,\gamma ) = \min \left\{ {\delta (\alpha ),\delta (\beta ), { 1 \over \gamma }\mathop {\max }\limits_{\sigma \in [\alpha \gamma ,\beta ]} \delta (\sigma )} \right\},$$ where δ is the entropy function δ(x)=xlog2x(1x)log2(1x).$$\delta (x) = - x\log _2 x - (1 - x)\;\log _2 (1 - x).$$

Keywords

  • Sequences of integers
  • lower asymptotic density
  • upper asymptotic density
  • gap density
  • Hausdorff dimension

MSC 2010

  • 11B05
  • 11K55
access type Open Access

Pair Correlations and Random Walks on the Integers

Published Online: 13 Jan 2017
Page range: 159 - 164

Abstract

Abstract

The paper gives conditions for a sequence of fractional parts of real numbers ({anx})n=1$\left( {\{ a_n x\} } \right)_{n = 1}^\infty $ to satisfy a pair correlation estimate. Here x is a fixed nonzero real number and (an)n=1$\left( {a_n } \right)_{n = 1}^\infty $ is a random walk on the integers.

Keywords

  • pair correlation
  • random walks

MSC 2010

  • 11K38
  • 60G50
access type Open Access

On Small Sets of Distribution Functions of Ratio Block Sequences

Published Online: 13 Jan 2017
Page range: 165 - 174

Abstract

Abstract

There are various methods how to describe and characterize distribution of elements of sets of positive integers. One of the most interesting is that using the set of all distribution functions of the corresponding ratio block sequence introduced in [Strauch, O.—Tóth, J.T.: Publ. Math. Debrecen 58 (2001), no. 4, 751–778]. In the present paper we give some sufficient conditions under which this set is small in a metric sense. As a corollary we obtain a new characterization of the case of asymptotic distribution.

Keywords

  • asymptotic distribution function
  • uniform distribution
  • block sequence

MSC 2010

  • 11B05
access type Open Access

Linear Recursive Odometers and Beta-Expansions

Published Online: 13 Jan 2017
Page range: 175 - 186

Abstract

Abstract

The aim of this paper is to study the connection between different properties related to β-expansions. In particular, the relation between two conditions, both ensuring purely discrete spectrum of the odometer, is analyzed. The first one is the so-called Hypothesis B for the G-odometers and the second one is denoted by (QM) and it has been introduced in the framework of tilings associated to Pisot β-numerations.

Keywords

  • beta-expansions
  • odometer
  • purely discrete spectrum
  • finiteness property

MSC 2010

  • 11A63
  • 11K16
  • 11B37
  • 28D05

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