<abstract xmlns="http://www.w3.org/1999/xhtml">In 2009, Grant, Shallit, and Stoll [Acta Arith. <bold>140</bold> (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.</abstract>

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

<abstract xmlns="http://www.w3.org/1999/xhtml">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 <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>Z</mi></mrow><mrow><msub><mrow><mi>B</mi></mrow><mn>2</mn></msub><mo>,</mo><mi>ν</mi></mrow><mrow><mi>κ</mi><mo>,</mo><mi>μ</mi></mrow></msubsup></mrow></math><tex-math>Z_{{{\rm B}_2},\nu }^{\kappa ,\mu }</tex-math></alternatives></inline-formula> 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 <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>Z</mi></mrow><mrow><msub><mrow><mi>B</mi></mrow><mn>2</mn></msub><mo>,</mo><mi>ν</mi></mrow><mrow><mi>κ</mi><mo>,</mo><mi>μ</mi></mrow></msubsup></mrow></math><tex-math>Z_{{{\rm B}_2},\nu }^{\kappa ,\mu }</tex-math></alternatives></inline-formula> is shown. If α1 = α2, then the following holds: if 1 &lt; α2 &lt; 2 the exact order is 𝒪 (<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><msqrt><mrow><mo>log</mo><mi>N</mi></mrow></msqrt></mrow><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>ε</mi></mrow></msup></mrow></mfrac></mrow></math><tex-math>{{\sqrt {\log N} } \over {{N^{1 - \varepsilon }}}}</tex-math></alternatives></inline-formula>) for some ε > 0, if α2 = 2 the exact order is 𝒪 (<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><msqrt><mrow><mo>log</mo><mi>N</mi></mrow></msqrt></mrow><mi>N</mi></mfrac></mrow></math><tex-math>{{\sqrt {\log N} } \over N}</tex-math></alternatives></inline-formula>) and if α2 > 2 the exact order is 𝒪 (<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_005.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><msqrt><mrow><mo>log</mo><mi>N</mi></mrow></msqrt></mrow><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>ε</mi></mrow></msup></mrow></mfrac></mrow></math><tex-math>{{\sqrt {\log N} } \over {{N^{1 + \varepsilon }}}}</tex-math></alternatives></inline-formula>) for some ε > 0. If α1 > α2, then the following holds: if 1 &lt; α2 &lt; 2 the exact order is 𝒪 (<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_006.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><msqrt><mrow><mo>log</mo><mi>N</mi></mrow></msqrt></mrow><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>ε</mi></mrow></msup></mrow></mfrac></mrow></math><tex-math>{{\sqrt {\log N} } \over {{N^{1 - \varepsilon }}}}</tex-math></alternatives></inline-formula>) for some ε > 0, if α2 = 2 the exact order is 𝒪 (<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_007.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mi>N</mi></mfrac></mrow></math><tex-math>{1 \over N}</tex-math></alternatives></inline-formula>) and if α2 > 2 the exact order is 𝒪 (<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_008.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><msqrt><mrow><mo>log</mo><mi>N</mi></mrow></msqrt></mrow><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>ε</mi></mrow></msup></mrow></mfrac></mrow></math><tex-math>{{\sqrt {\log N} } \over {{N^{1 + \varepsilon }}}}</tex-math></alternatives></inline-formula>) for some ε > 0. Here N = Bν, where Bν denotes the number of the points of the nets <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_009.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>Z</mi></mrow><mrow><msub><mrow><mi>B</mi></mrow><mn>2</mn></msub><mo>,</mo><mi>ν</mi></mrow><mrow><mi>κ</mi><mo>,</mo><mi>μ</mi></mrow></msubsup></mrow></math><tex-math>Z_{{{\rm B}_2},\nu }^{\kappa ,\mu }</tex-math></alternatives></inline-formula>.</abstract>

South West University “Neofit Rilski”

On the (Vil<italic>B</italic>2; <italic>α</italic>; <italic>γ</italic>)-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System

<abstract xmlns="http://www.w3.org/1999/xhtml">We exhibit a class of Littlewood polynomials that are not <italic>L</italic><italic>α</italic>-flat for any <italic>α</italic> ≥ 0. Indeed, it is shown that the sequence of Littlewood polynomials is not <italic>L</italic><italic>α</italic>-flat, <italic>α</italic> ≥ 0, when the frequency of −1 is not in the interval ]<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0003_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math><tex-math>{1 \over 4}</tex-math></alternatives></inline-formula>, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0003_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></math><tex-math>{3 \over 4}</tex-math></alternatives></inline-formula>[ We further obtain a generalization of Jensen-Jensen-Hoholdt’s result by establishing that the sequence of Littlewood polynomials is not <italic>L</italic><italic>α</italic>-flat for any <italic>α></italic> 2 if the frequency of <italic>−</italic>1 is not <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0003_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math><tex-math>{1 \over 2}</tex-math></alternatives></inline-formula>. Finally, we prove that the sequence of palindromic Littlewood polynomials with even degrees are not <italic>L</italic><italic>α</italic>-flat for any <italic>α ≥</italic> 0, and we provide a lemma on the existence of <italic>c</italic>-flat polynomials.</abstract>

A Class of Littlewood Polynomials that are Not <italic>L</italic><italic>α</italic>-Flat

<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>K</italic> be a number field, and let <italic>G</italic> be a finitely generated and torsion-free subgroup of <italic>K</italic><italic>×</italic>. For almost all primes p of <italic>K</italic>, we consider the order of the cyclic group (<italic>G</italic> 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 ℓ<italic>e</italic> is a prime power and <italic>a</italic> is a multiple of ℓ (and <italic>a</italic> is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of <italic>K</italic> such that the order of (<italic>G</italic> mod 𝔭) is congruent to <italic>a</italic> modulo ℓ<italic>e</italic> only depends on <italic>a</italic> through its ℓ-adic valuation.</abstract>

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

<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>f</italic> (<italic>x</italic>) bea monicpolynomialwith integer coefficients and integers <italic>r</italic>1,..., <italic>r</italic><italic>n</italic> with 0 ≤ <italic>r</italic>1 ≤··· ≤ <italic>r</italic><italic>n &lt;p</italic> the <italic>n</italic> roots of <italic>f</italic> (<italic>x</italic>) ≡ 0mod <italic>p</italic> for a prime <italic>p</italic>. We proposed conjectures on the distribution of the point (<italic>r</italic>1<italic>/p,...,r</italic><italic>n</italic><italic>/p</italic>) in the previous papers. One aim of this paper is to revise them for a reducible polynomial <italic>f</italic> (<italic>x</italic>), and the other is to show that they imply the one-dimensional equidistribution of <italic>r</italic>1<italic>/p,...,r</italic><italic>n</italic><italic>/p</italic> for an irreducible polynomial <italic>f</italic> (<italic>x</italic>) by a geometric way.</abstract>

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

<abstract xmlns="http://www.w3.org/1999/xhtml">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.</abstract>

Quantization for a Mixture of Uniform Distributions Associated with Probability Vectors

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Uniform distribution theory

 The journal is published by:The BOKU-University of Natural Resources and Applied Life Sciences (Vienna, Austria) and Institute of Mathematics of the Slovak Academy of Sciences, (Bratislava, Slovakia)  The aim of this journal is to publish original research papers on various aspects of uniform distribution theory, especially theoretical and computational aspects of combinatorial, diophantine and probabilistic number theory. In particular we welcome submissions on the following topics:   Distribution of one dimensional and multidimensional sequences.  Distribution of binary sequences.  Distribution of integer sequences and sequences from groups and  Distribution problems in finite abstract sets.  Continuously uniform distributions.  Discrepancies.  Pseudo-random number generators.  Quasi-Monte Carlo integration.  Quasi-Monte Carlo methods in financial mathematics.  Theory of densities.  Theory of distribution functions of sequences.  Distribution of integer points in large domains.  Spectral properties of sequences.  Dynamics emerging from sequences.  Number theoretic ciphers and codes.  Combinatorial number theory.  Trigonometric sums.  Diophantine approximations and diophantine equations.  Continued fractions.  Quantum chaos.  In terms of the AMS subject classification, the main subject areas supported are:11K, 11J, 11B and 11L.  Company Registration Number: 00166791; ISSN 1336–913X; EV 4556/12; math.boku.ac.at/udt  Archiving  Sciendo archives the contents of this journal in Portico - digital long-term preservation service of scholarly books, journals and collections.  Plagiarism Policy  The editorial board is participating in a growing community of Similarity Check System's users in order to ensure that the content published is original and trustworthy. Similarity Check is a medium that allows for comprehensive manuscripts screening, aimed to eliminate plagiarism and provide a high standard and quality peer-review process. 

Uniform distribution theory

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

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

On the (Vil_B₂; α; γ)-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System

A Class of Littlewood Polynomials that are Not L^α-Flat

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

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

Quantization for a Mixture of Uniform Distributions Associated with Probability Vectors

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All Volumes and Issues in this Journal

Uniform distribution theory

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

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

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

A Class of Littlewood Polynomials that are Not Lα-Flat

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

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

Quantization for a Mixture of Uniform Distributions Associated with Probability Vectors

Search

All Volumes and Issues in this Journal

On the (Vil_B₂; α; γ)-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System

A Class of Littlewood Polynomials that are Not L^α-Flat