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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 12 (2017): Issue 2 (December 2017)

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

Search

9 Articles
access type Open Access

Distribution Functions for Subsequences of Generalized Van Der Corput Sequences

Published Online: 30 Jan 2018
Page range: 1 - 10

Abstract

Abstract

For an integer b > 1 let (φb(n))n≥0 denote the van der Corput sequence base in b in [0, 1). Answering a question of O. Strauch, C. Aistleitner and M. Hofer showed that the distribution function of (φb(n), φb(n + 1), . . . , φb(n + s − 1))n≥0 on [0, 1)s exists and is a copula. The first and third authors of the present paper showed that this phenomenon extends to a broad class of subsequences of the van der Corput sequence. In this result we extend this paper still further and show that this phenomenon is also true for more general numeration systems based on the beta expansion of W. Parry and A. Rényi.

Keywords

  • Generalised van der Corput sequences
  • beta-expansions
  • Hartman distributed sequences of integers
  • distribution functions

MSC 2010

  • 11K31
  • 40A05
access type Open Access

Upper Bounds for Double Exponential Sums Along a Subsequence

Published Online: 30 Jan 2018
Page range: 11 - 24

Abstract

Abstract

We consider a class of double exponential sums studied in a paper of Sinai and Ulcigrai. They proved a linear bound for these sums along the sequence of denominators in the continued fraction expansion of α, provided α is badly-approximable. We provide a proof of a result, which includes a simple proof of their theorem, and which applies for all irrational α.

Keywords

  • continued fraction
  • badly-approximable α
  • double-exponential sum
  • discrepancy
  • Koksma-Hlawka inequality
  • Ostrowski expansion

MSC 2010

  • 11J70
  • 11L03
  • 11L07
access type Open Access

Palindromic Closures and Thue-Morse Substitution for Markoff Numbers

Published Online: 30 Jan 2018
Page range: 25 - 35

Abstract

Abstract

We state a new formula to compute the Markoff numbers using iterated palindromic closure and the Thue-Morse substitution. The main theorem shows that for each Markoff number m, there exists a word v ∈ {a, b}∗ such that m − 2 is equal to the length of the iterated palindromic closure of the iterated antipalindromic closure of the word av. This construction gives a new recursive construction of the Markoff numbers by the lengths of the words involved in the palindromic closure. This construction interpolates between the Fibonacci numbers and the Pell numbers.

Keywords

  • iterated palindromic closure
  • Thue-Morse Substitution
  • Markoff spectra

MSC 2010

  • 68R15
  • 52C99
access type Open Access

Discrepancy of Generalized LS-Sequences

Published Online: 30 Jan 2018
Page range: 37 - 63

Abstract

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.

Keywords

  • Discrepancy
  • LS-sequence
  • uniform distribution
  • beta-expansion

MSC 2010

  • 11K38
  • 11J71
  • 11A67
access type Open Access

Uncanny Subsequence Selections That Generate Normal Numbers

Published Online: 30 Jan 2018
Page range: 65 - 75

Abstract

Abstract

Given a real number 0.a1a2a3 . . . that is normal to base b, we examine increasing sequences ni so that the number 0.an1an2an3 . . . are normal to base b. Classically, it is known that if the ni form an arithmetic progression, then this will work. We give several more constructions including ni that are recursively defined based on the digits ai. Of particular interest, we show that if a number is normal to base b, then removing all the digits from its expansion which equal (b−1) leaves a base-(b−1) expansion that is normal to base (b − 1)

Keywords

  • normal numbers

MSC 2010

  • 11K16
access type Open Access

On the Closure of the Image of the Generalized Divisor Function

Published Online: 30 Jan 2018
Page range: 77 - 90

Abstract

Abstract

For any real number s, let σs be the generalized divisor function, i.e., the arithmetic function defined by σs(n) := ∑d|n ds, for all positive integers n. We prove that for any r > 1 the topological closure of σ−r(N+) is the union of a finite number of pairwise disjoint closed intervals I1, . . . , I. Moreover, for k = 1, . . . , ℓ, we show that the set of positive integers n such that σ−r(n) ∈ Ik has a positive rational asymptotic density dk. In fact, we provide a method to give exact closed form expressions for I1, . . . , I and d1, . . . , d, assuming to know r with sufficient precision. As an example, we show that for r = 2 it results ℓ = 3, I1 = [1, π2/9], I2 = [10/9, π2/8], I3 = [5/4, π2/6], d1 = 1/3, d2 = 1/6, and d3 = 1/2.

Keywords

  • Arithmetic functions
  • sum of divisors
  • topological closure
  • asymptotic densities

MSC 2010

  • 11A25
  • 11N37
  • 11N64
  • 11Y99
access type Open Access

Notes on the Distribution of Roots Modulo a Prime of a Polynomial

Published Online: 30 Jan 2018
Page range: 91 - 117

Abstract

Abstract

Let f(x) be a monic polynomial in Z[x] with roots α1, . . ., αn. We point out the importance of linear relations among 1, α1, . . . , αn over rationals with respect to the distribution of local roots of f modulo a prime. We formulate it as a conjectural uniform distribution in some sense, which elucidates data in previous papers.

Keywords

  • distribution
  • polynomial
  • roots modulo a prime

MSC 2010

  • 11K
access type Open Access

Corrigendum to the h-Critical Number of Finite Abelian Groups

Published Online: 30 Jan 2018
Page range: 119 - 124

Abstract

Abstract

We here correct two errors of our paper cited in the title: one in the statement of Theorem 5 and another in the proof of Theorem 11.

Keywords

  • critical number
  • abelian groups
  • sumsets
  • restricted sumsets

MSC 2010

  • Primary 11B75
  • Secondary 05D99
  • 11B25
  • 11P70
  • 20K01
access type Open Access

A Note on the Continued Fraction of Minkowski

Published Online: 30 Jan 2018
Page range: 125 - 130

Abstract

Abstract

Denote by Θ12, · · · the sequence of approximation coefficients of Minkowski’s diagonal continued fraction expansion of a real irrational number x. For almost all x this is a uniformly distributed sequence in the interval [0, 1/2 ]. The average distance between two consecutive terms of this sequence and their correlation coefficient are explicitly calculated and it is shown why these two values are close to 1/6 and 0, respectively, the corresponding values for a random sequence in [0, 1/2].

Keywords

  • Continued fractions
  • approximation coefficients
  • metrical theory

MSC 2010

  • 11K50
9 Articles
access type Open Access

Distribution Functions for Subsequences of Generalized Van Der Corput Sequences

Published Online: 30 Jan 2018
Page range: 1 - 10

Abstract

Abstract

For an integer b > 1 let (φb(n))n≥0 denote the van der Corput sequence base in b in [0, 1). Answering a question of O. Strauch, C. Aistleitner and M. Hofer showed that the distribution function of (φb(n), φb(n + 1), . . . , φb(n + s − 1))n≥0 on [0, 1)s exists and is a copula. The first and third authors of the present paper showed that this phenomenon extends to a broad class of subsequences of the van der Corput sequence. In this result we extend this paper still further and show that this phenomenon is also true for more general numeration systems based on the beta expansion of W. Parry and A. Rényi.

Keywords

  • Generalised van der Corput sequences
  • beta-expansions
  • Hartman distributed sequences of integers
  • distribution functions

MSC 2010

  • 11K31
  • 40A05
access type Open Access

Upper Bounds for Double Exponential Sums Along a Subsequence

Published Online: 30 Jan 2018
Page range: 11 - 24

Abstract

Abstract

We consider a class of double exponential sums studied in a paper of Sinai and Ulcigrai. They proved a linear bound for these sums along the sequence of denominators in the continued fraction expansion of α, provided α is badly-approximable. We provide a proof of a result, which includes a simple proof of their theorem, and which applies for all irrational α.

Keywords

  • continued fraction
  • badly-approximable α
  • double-exponential sum
  • discrepancy
  • Koksma-Hlawka inequality
  • Ostrowski expansion

MSC 2010

  • 11J70
  • 11L03
  • 11L07
access type Open Access

Palindromic Closures and Thue-Morse Substitution for Markoff Numbers

Published Online: 30 Jan 2018
Page range: 25 - 35

Abstract

Abstract

We state a new formula to compute the Markoff numbers using iterated palindromic closure and the Thue-Morse substitution. The main theorem shows that for each Markoff number m, there exists a word v ∈ {a, b}∗ such that m − 2 is equal to the length of the iterated palindromic closure of the iterated antipalindromic closure of the word av. This construction gives a new recursive construction of the Markoff numbers by the lengths of the words involved in the palindromic closure. This construction interpolates between the Fibonacci numbers and the Pell numbers.

Keywords

  • iterated palindromic closure
  • Thue-Morse Substitution
  • Markoff spectra

MSC 2010

  • 68R15
  • 52C99
access type Open Access

Discrepancy of Generalized LS-Sequences

Published Online: 30 Jan 2018
Page range: 37 - 63

Abstract

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.

Keywords

  • Discrepancy
  • LS-sequence
  • uniform distribution
  • beta-expansion

MSC 2010

  • 11K38
  • 11J71
  • 11A67
access type Open Access

Uncanny Subsequence Selections That Generate Normal Numbers

Published Online: 30 Jan 2018
Page range: 65 - 75

Abstract

Abstract

Given a real number 0.a1a2a3 . . . that is normal to base b, we examine increasing sequences ni so that the number 0.an1an2an3 . . . are normal to base b. Classically, it is known that if the ni form an arithmetic progression, then this will work. We give several more constructions including ni that are recursively defined based on the digits ai. Of particular interest, we show that if a number is normal to base b, then removing all the digits from its expansion which equal (b−1) leaves a base-(b−1) expansion that is normal to base (b − 1)

Keywords

  • normal numbers

MSC 2010

  • 11K16
access type Open Access

On the Closure of the Image of the Generalized Divisor Function

Published Online: 30 Jan 2018
Page range: 77 - 90

Abstract

Abstract

For any real number s, let σs be the generalized divisor function, i.e., the arithmetic function defined by σs(n) := ∑d|n ds, for all positive integers n. We prove that for any r > 1 the topological closure of σ−r(N+) is the union of a finite number of pairwise disjoint closed intervals I1, . . . , I. Moreover, for k = 1, . . . , ℓ, we show that the set of positive integers n such that σ−r(n) ∈ Ik has a positive rational asymptotic density dk. In fact, we provide a method to give exact closed form expressions for I1, . . . , I and d1, . . . , d, assuming to know r with sufficient precision. As an example, we show that for r = 2 it results ℓ = 3, I1 = [1, π2/9], I2 = [10/9, π2/8], I3 = [5/4, π2/6], d1 = 1/3, d2 = 1/6, and d3 = 1/2.

Keywords

  • Arithmetic functions
  • sum of divisors
  • topological closure
  • asymptotic densities

MSC 2010

  • 11A25
  • 11N37
  • 11N64
  • 11Y99
access type Open Access

Notes on the Distribution of Roots Modulo a Prime of a Polynomial

Published Online: 30 Jan 2018
Page range: 91 - 117

Abstract

Abstract

Let f(x) be a monic polynomial in Z[x] with roots α1, . . ., αn. We point out the importance of linear relations among 1, α1, . . . , αn over rationals with respect to the distribution of local roots of f modulo a prime. We formulate it as a conjectural uniform distribution in some sense, which elucidates data in previous papers.

Keywords

  • distribution
  • polynomial
  • roots modulo a prime

MSC 2010

  • 11K
access type Open Access

Corrigendum to the h-Critical Number of Finite Abelian Groups

Published Online: 30 Jan 2018
Page range: 119 - 124

Abstract

Abstract

We here correct two errors of our paper cited in the title: one in the statement of Theorem 5 and another in the proof of Theorem 11.

Keywords

  • critical number
  • abelian groups
  • sumsets
  • restricted sumsets

MSC 2010

  • Primary 11B75
  • Secondary 05D99
  • 11B25
  • 11P70
  • 20K01
access type Open Access

A Note on the Continued Fraction of Minkowski

Published Online: 30 Jan 2018
Page range: 125 - 130

Abstract

Abstract

Denote by Θ12, · · · the sequence of approximation coefficients of Minkowski’s diagonal continued fraction expansion of a real irrational number x. For almost all x this is a uniformly distributed sequence in the interval [0, 1/2 ]. The average distance between two consecutive terms of this sequence and their correlation coefficient are explicitly calculated and it is shown why these two values are close to 1/6 and 0, respectively, the corresponding values for a random sequence in [0, 1/2].

Keywords

  • Continued fractions
  • approximation coefficients
  • metrical theory

MSC 2010

  • 11K50

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