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

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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

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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 15 (2020): Issue 1 (June 2020)

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

Search

6 Articles
access type Open Access

Discrete Correlation of Order 2 of Generalized Rudin-Shapiro Sequences on Alphabets of Arbitrary Size

Published Online: 24 Jul 2020
Page range: 1 - 26

Abstract

Abstract

In 2009, Grant, Shallit, and Stoll [Acta Arith. 140 (2009), [345–368] constructed a large family of pseudorandom sequences, called generalized Rudin--Shapiro sequences, for which they established some results about the average of discrete correlation coefficients of order 2 in cases where the size of the alphabet is a prime number or a squarefree product of primes. We establish similar results for an even larger family of pseudorandom sequences, constructed via difference matrices, in the case of an alphabet of any size. The constructions generalize those from Grant et al. In the case where the size of the alphabet is squarefree and where there are at least two prime factors, we obtain an improvement in the error term by comparison with the result of Grant et al.

Keywords

  • discrete correlation
  • Rudin-Shapiro sequence
  • difference matrix
  • exponential sums

MSC 2010

  • 11A63
  • 11K31
  • 68R15
access type Open Access

On the (VilB2; α; γ)-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System

Published Online: 24 Jul 2020
Page range: 27 - 50

Abstract

Abstract

In the present paper the so-called (VilBs; α; γ)-diaphony as a quantitative measure for the distribution of sequences and nets is considered. A class of two-dimensional nets ZB2,νκ,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu } of type of Zaremba-Halton constructed in a generalized B2-adic system or Cantor system is introduced and the (VilB2; α; γ)-diaphony of these nets is studied. The influence of the vector α = (α1, α2) of exponential parameters to the exact order of the (VilB2; α; γ)-diaphony of the nets ZB2,νκ,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu } is shown. If α1 = α2, then the following holds: if 1 < α2 < 2 the exact order is 𝒪 (logNN1-ε{{\sqrt {\log N} } \over {{N^{1 - \varepsilon }}}}) for some ε > 0, if α2 = 2 the exact order is 𝒪 (logNN{{\sqrt {\log N} } \over N}) and if α2 > 2 the exact order is 𝒪 (logNN1+ε{{\sqrt {\log N} } \over {{N^{1 + \varepsilon }}}}) for some ε > 0. If α1 > α2, then the following holds: if 1 < α2 < 2 the exact order is 𝒪 (logNN1-ε{{\sqrt {\log N} } \over {{N^{1 - \varepsilon }}}}) for some ε > 0, if α2 = 2 the exact order is 𝒪 (1N{1 \over N}) and if α2 > 2 the exact order is 𝒪 (logNN1+ε{{\sqrt {\log N} } \over {{N^{1 + \varepsilon }}}}) for some ε > 0. Here N = Bν, where Bν denotes the number of the points of the nets ZB2,νκ,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu }.

Keywords

  • Diaphony
  • Vilenkin function
  • Walsh function
  • nets of type of Zaremba-Halton
  • van der Corput sequence

MSC 2010

  • 11K06
  • 11K31
  • 11K36
  • 65C05
access type Open Access

A Class of Littlewood Polynomials that are Not Lα-Flat

Published Online: 24 Jul 2020
Page range: 51 - 74

Abstract

Abstract

We exhibit a class of Littlewood polynomials that are not Lα-flat for any α ≥ 0. Indeed, it is shown that the sequence of Littlewood polynomials is not Lα-flat, α ≥ 0, when the frequency of −1 is not in the interval ]14{1 \over 4}, 34{3 \over 4}[ We further obtain a generalization of Jensen-Jensen-Hoholdt’s result by establishing that the sequence of Littlewood polynomials is not Lα-flat for any α> 2 if the frequency of 1 is not 12{1 \over 2}. Finally, we prove that the sequence of palindromic Littlewood polynomials with even degrees are not Lα-flat for any α ≥ 0, and we provide a lemma on the existence of c-flat polynomials.

Keywords

  • Merit factor
  • flat polynomials
  • ultraflat polynomials
  • Erd˝ os-Newman flatness problem
  • Littlewood flatness problem
  • digital transmission
  • palindromic polynomial
  • TurynGolay’s conjecture

MSC 2010

  • Primary 42A05, 42A55
  • Secondary 37A05, 37A30
access type Open Access

Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II

Published Online: 24 Jul 2020
Page range: 75 - 92

Abstract

Abstract

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K×. For almost all primes p of K, we consider the order of the cyclic group (G mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓe is a prime power and a is a multiple of ℓ (and a is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of K such that the order of (G mod 𝔭) is congruent to a modulo ℓe only depends on a through its ℓ-adic valuation.

Keywords

  • Number field
  • reduction
  • multiplicative order
  • arithmetic progression
  • density

MSC 2010

  • Primary: 11R44
  • Secondary: 11R45,11R18, 11R21
access type Open Access

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

Published Online: 24 Jul 2020
Page range: 93 - 104

Abstract

Abstract

Let f (x) bea monicpolynomialwith integer coefficients and integers r1,..., rn with 0 ≤ r1 ≤··· ≤ rn <p the n roots of f (x) ≡ 0mod p for a prime p. We proposed conjectures on the distribution of the point (r1/p,...,rn/p) in the previous papers. One aim of this paper is to revise them for a reducible polynomial f (x), and the other is to show that they imply the one-dimensional equidistribution of r1/p,...,rn/p for an irreducible polynomial f (x) by a geometric way.

Keywords

  • Equidistribution
  • polynomial
  • roots modulo a prime

MSC 2010

  • 11K
access type Open Access

Quantization for a Mixture of Uniform Distributions Associated with Probability Vectors

Published Online: 24 Jul 2020
Page range: 105 - 142

Abstract

Abstract

The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability distribution by a discrete distribution. Mixtures of probability distributions, also known as mixed distributions, are an exciting new area for optimal quantization. In this paper, we investigate the optimal quantization for three different mixed distributions generated by uniform distributions associated with probability vectors.

Keywords

  • Mixed distribution
  • uniform distribution
  • optimal sets
  • quantization error
  • quantization dimension
  • quantization coefficient

MSC 2010

  • 60Exx
  • 94A34
6 Articles
access type Open Access

Discrete Correlation of Order 2 of Generalized Rudin-Shapiro Sequences on Alphabets of Arbitrary Size

Published Online: 24 Jul 2020
Page range: 1 - 26

Abstract

Abstract

In 2009, Grant, Shallit, and Stoll [Acta Arith. 140 (2009), [345–368] constructed a large family of pseudorandom sequences, called generalized Rudin--Shapiro sequences, for which they established some results about the average of discrete correlation coefficients of order 2 in cases where the size of the alphabet is a prime number or a squarefree product of primes. We establish similar results for an even larger family of pseudorandom sequences, constructed via difference matrices, in the case of an alphabet of any size. The constructions generalize those from Grant et al. In the case where the size of the alphabet is squarefree and where there are at least two prime factors, we obtain an improvement in the error term by comparison with the result of Grant et al.

Keywords

  • discrete correlation
  • Rudin-Shapiro sequence
  • difference matrix
  • exponential sums

MSC 2010

  • 11A63
  • 11K31
  • 68R15
access type Open Access

On the (VilB2; α; γ)-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System

Published Online: 24 Jul 2020
Page range: 27 - 50

Abstract

Abstract

In the present paper the so-called (VilBs; α; γ)-diaphony as a quantitative measure for the distribution of sequences and nets is considered. A class of two-dimensional nets ZB2,νκ,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu } of type of Zaremba-Halton constructed in a generalized B2-adic system or Cantor system is introduced and the (VilB2; α; γ)-diaphony of these nets is studied. The influence of the vector α = (α1, α2) of exponential parameters to the exact order of the (VilB2; α; γ)-diaphony of the nets ZB2,νκ,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu } is shown. If α1 = α2, then the following holds: if 1 < α2 < 2 the exact order is 𝒪 (logNN1-ε{{\sqrt {\log N} } \over {{N^{1 - \varepsilon }}}}) for some ε > 0, if α2 = 2 the exact order is 𝒪 (logNN{{\sqrt {\log N} } \over N}) and if α2 > 2 the exact order is 𝒪 (logNN1+ε{{\sqrt {\log N} } \over {{N^{1 + \varepsilon }}}}) for some ε > 0. If α1 > α2, then the following holds: if 1 < α2 < 2 the exact order is 𝒪 (logNN1-ε{{\sqrt {\log N} } \over {{N^{1 - \varepsilon }}}}) for some ε > 0, if α2 = 2 the exact order is 𝒪 (1N{1 \over N}) and if α2 > 2 the exact order is 𝒪 (logNN1+ε{{\sqrt {\log N} } \over {{N^{1 + \varepsilon }}}}) for some ε > 0. Here N = Bν, where Bν denotes the number of the points of the nets ZB2,νκ,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu }.

Keywords

  • Diaphony
  • Vilenkin function
  • Walsh function
  • nets of type of Zaremba-Halton
  • van der Corput sequence

MSC 2010

  • 11K06
  • 11K31
  • 11K36
  • 65C05
access type Open Access

A Class of Littlewood Polynomials that are Not Lα-Flat

Published Online: 24 Jul 2020
Page range: 51 - 74

Abstract

Abstract

We exhibit a class of Littlewood polynomials that are not Lα-flat for any α ≥ 0. Indeed, it is shown that the sequence of Littlewood polynomials is not Lα-flat, α ≥ 0, when the frequency of −1 is not in the interval ]14{1 \over 4}, 34{3 \over 4}[ We further obtain a generalization of Jensen-Jensen-Hoholdt’s result by establishing that the sequence of Littlewood polynomials is not Lα-flat for any α> 2 if the frequency of 1 is not 12{1 \over 2}. Finally, we prove that the sequence of palindromic Littlewood polynomials with even degrees are not Lα-flat for any α ≥ 0, and we provide a lemma on the existence of c-flat polynomials.

Keywords

  • Merit factor
  • flat polynomials
  • ultraflat polynomials
  • Erd˝ os-Newman flatness problem
  • Littlewood flatness problem
  • digital transmission
  • palindromic polynomial
  • TurynGolay’s conjecture

MSC 2010

  • Primary 42A05, 42A55
  • Secondary 37A05, 37A30
access type Open Access

Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II

Published Online: 24 Jul 2020
Page range: 75 - 92

Abstract

Abstract

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K×. For almost all primes p of K, we consider the order of the cyclic group (G mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓe is a prime power and a is a multiple of ℓ (and a is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of K such that the order of (G mod 𝔭) is congruent to a modulo ℓe only depends on a through its ℓ-adic valuation.

Keywords

  • Number field
  • reduction
  • multiplicative order
  • arithmetic progression
  • density

MSC 2010

  • Primary: 11R44
  • Secondary: 11R45,11R18, 11R21
access type Open Access

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

Published Online: 24 Jul 2020
Page range: 93 - 104

Abstract

Abstract

Let f (x) bea monicpolynomialwith integer coefficients and integers r1,..., rn with 0 ≤ r1 ≤··· ≤ rn <p the n roots of f (x) ≡ 0mod p for a prime p. We proposed conjectures on the distribution of the point (r1/p,...,rn/p) in the previous papers. One aim of this paper is to revise them for a reducible polynomial f (x), and the other is to show that they imply the one-dimensional equidistribution of r1/p,...,rn/p for an irreducible polynomial f (x) by a geometric way.

Keywords

  • Equidistribution
  • polynomial
  • roots modulo a prime

MSC 2010

  • 11K
access type Open Access

Quantization for a Mixture of Uniform Distributions Associated with Probability Vectors

Published Online: 24 Jul 2020
Page range: 105 - 142

Abstract

Abstract

The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability distribution by a discrete distribution. Mixtures of probability distributions, also known as mixed distributions, are an exciting new area for optimal quantization. In this paper, we investigate the optimal quantization for three different mixed distributions generated by uniform distributions associated with probability vectors.

Keywords

  • Mixed distribution
  • uniform distribution
  • optimal sets
  • quantization error
  • quantization dimension
  • quantization coefficient

MSC 2010

  • 60Exx
  • 94A34

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