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

In 1895, Vahlen proved a theorem concerning a simultaneous approximation of a real number by its two consecutive convergent. In this paper, we will provide a sharper bound for the theorem.
</abstract>

Refinement of the Theorem of Vahlen

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

For polynomials <italic>f</italic> (<italic>x</italic>),<italic>g</italic>1(<italic>x</italic>),<italic>g</italic>2(<italic>x</italic>)over ℤ, we report several observations about the density of primes <italic>p</italic> for which <italic>f</italic> (<italic>x</italic>) is fully splitting at <italic>p</italic> and <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2023-0002_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mrow><mo>{</mo> <mrow><mfrac><mrow><msub><mrow><mi>g</mi></mrow><mn>1</mn></msub><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow><mi>p</mi></mfrac></mrow> <mo>}</mo></mrow><mo>&lt;</mo><mrow><mo>{</mo> <mrow><mfrac><mrow><msub><mrow><mi>g</mi></mrow><mn>2</mn></msub><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow><mi>p</mi></mfrac></mrow> <mo>}</mo></mrow></mrow></math>
<tex-math>\left\{ {{{{g_1}\left( r \right)} \over p}} \right\} &lt; \left\{ {{{{g_2}\left( r \right)} \over p}} \right\}</tex-math>
</alternatives>
</inline-formula> for some root <italic>r</italic> of <italic>f</italic> (<italic>x</italic>) ≡ 0 mod <italic>p</italic>.
</abstract>

Notes on the Distribution of Roots Modulo Primes of a Polynomial IV

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

For a probability measure <italic>μ</italic> on [0, 1] without discrete component, the best possible order of approximation by a finite point set in terms of the star-discrepancy is &amp;inline as has been proven relatively recently. However, if <italic>μ</italic> contains a discrete component no non-trivial lower bound holds in general because it is straightforward to construct examples without any approximation error in this case. This might explain, why the approximation of discrete measures on [0, 1] by finite point sets has so far not been completely covered in the existing literature. In this note, we close the gap by giving a complete description for discrete measures. Most importantly, we prove that for any discrete measures (not supported on one point only) the best possible order of approximation is for infinitely many <italic>N</italic> bounded from below by &amp;inline for some constant 6 ≥ <italic>c></italic> 2 which depends on the measure. This implies, that for a finitely supported discrete measure on [0, 1]d the known possible order of approximation &amp;inline is indeed the optimal one.
</abstract>

Approximation of Discrete Measures by Finite Point Sets

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

We give a necessary and sufficient condition for equidistribution of continuous functions along monotone compact covers on locally compact spaces. We show the existence of equidistributed mappings along Bohr nets arising from group actions. Using almost periodic means, we give an analogue of Weyl’s equidistribution criterion for continuous functions with values in arbitrary topological groups. We prove van der Corput’s inequality on the lattice ℕm for vectors in Hilbert spaces, and use this inequality to extend Hlawka’s equidistribution theorem to functions on the lattice ℕm (<italic>m</italic> ≥ 1) with values in arbitrary topological groups.
</abstract>

Equidistribution of Continuous Functions Along Monotone Compact Covers

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

For <italic>m, d</italic> ∈ ℕ, a jittered sample of <italic>N</italic> = <italic>m</italic>d points can be constructed by partitioning [0, 1]d into <italic>m</italic>d axis-aligned equivolume boxes and placing one point independently and uniformly at random inside each box. We utilise a formula for the expected ℒ2−discrepancy of stratified samples stemming from general equivolume partitions of [0, 1]d which recently appeared, to derive a closed form expression for the expected ℒ2−discrepancy of a jittered point set for any <italic>m, d</italic> ∈ ℕ. As a second main result we derive a similar formula for the expected Hickernell ℒ2−discrepancy of a jittered point set which also takes all projections of the point set to lower dimensional faces of the unit cube into account.
</abstract>

On the Expected ℒ2–Discrepancy of Jittered Sampling

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

It is important in cryptographic applications that the “key” used should be generated from a random seed. Thus, if the Legendre symbol sequence generated by a polynomial (as proposed by Hoffstein and Lieman) is used, that is
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2023-0006_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow><mo>{</mo> <mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>f</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow><mi>p</mi></mfrac></mrow><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mi>f</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow><mi>p</mi></mfrac></mrow><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mi>f</mi><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow><mi>p</mi></mfrac></mrow><mo>)</mo></mrow><mo>,</mo><mo>⋯</mo><mo>,</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mi>f</mi><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow><mi>p</mi></mfrac></mrow><mo>)</mo></mrow></mrow> <mo>}</mo></mrow><mo>,</mo></mrow></math>
<tex-math>\left\{ {\left( {{{f\left( 1 \right)} \over p}} \right),\left( {{{f\left( 2 \right)} \over p}} \right),\left( {{{f\left( 3 \right)} \over p}} \right), \cdots ,\left( {{{f\left( p \right)} \over p}} \right)} \right\},</tex-math>
</alternatives>
</disp-formula>
then it is important to choose the polynomial <italic>f</italic> “almost” at random. Goubin, Mauduit, and Sárközy presented some not very restrictive conditions on the polynomial <italic>f</italic>, but these conditions may not be satisfied if we choose a “truly” random polynomial. However, how can it be guaranteed that the pseudorandom measures of the sequence should be small for almost "random" polynomials? These semirandom polynomials will be constructed with as few modifications as necessary from a truly random polynomial.
</abstract>

Random Polynomials in Legendre Symbol Sequences

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

In this paper, we study an efficient algorithm for constructing point sets underlying quasi-Monte Carlo integration rules for weighted Korobov classes. The algorithm presented is a reduced fast component-by-component digit-by-digit (CBC-DBD) algorithm, which is useful for situations where the weights in the function space show a sufficiently fast decay. The advantage of the algorithm presented here is that the computational effort can be independent of the dimension of the integration problem to be treated if suitable assumptions on the integrand are met. By considering a reduced digit-by-digit construction, we allow an integration algorithm to be less precise with respect to the number of bits in those components of the problem that are considered less important. The new reduced CBC-DBD algorithm is designed to work for the construction of lattice point sets, and the corresponding integration rules (so-called lattice rules) can be used to treat functions in different kinds of function spaces. We show that the integration rules constructed by our algorithm satisfy error bounds of almost optimal convergence order. Furthermore, we give details on an efficient implementation such that we obtain a considerable speed-up of a previously known CBC-DBD algorithm that has been studied in the paper <italic>Digit-by-digit and component-by-component constructions of lattice rules for periodic functions with unknown smoothness</italic> by Ebert, Kritzer, Nuyens, and Osisiogu, published in the Journal of Complexity in 2021. This improvement is illustrated by numerical results.
</abstract>

On a Reduced Component-by-Component Digit-by-Digit Construction of Lattice Point Sets

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

We prove an easy statement about inhomogeneous approximation for non-singular vectors in metric theory of Diophantine approximation.
</abstract>

A Note on Well Distributed Sequences

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

Two-dimensional distribution function <italic>g</italic>(<italic>x, y</italic>) defined in [0, 1]2 is called copula, if <italic>g</italic>(<italic>x,</italic> 1) = <italic>x</italic> and <italic>g</italic>(1,<italic>y</italic>)= <italic>y</italic> for every <italic>x, y</italic>. Similarly, <italic>s</italic>-dimensional copula is a distribution function <italic>g</italic>(<italic>x</italic>1,<italic>x</italic>2,...,<italic>x</italic>s) such that every <italic>k</italic>-dimensional face function
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2023-0009_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>g</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mn>1</mn></msub></mrow></msub><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mn>2</mn></msub></mrow></msub><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mi>k</mi></msub></mrow></msub><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math>
<tex-math>g\left( {1, \ldots ,1,{x_{{i_1}}},1, \ldots ,1,{x_{{i_2}}},1, \ldots ,1,{x_{{i_k}}},1, \ldots ,1} \right)</tex-math>
</alternatives>
</disp-formula>
is equal to <italic>x</italic>i1 <italic>x</italic>i2 ...<italic>x</italic>ik for some but fixed <italic>k</italic>. In this paper we summarize and extend all known parts of copulas.
In this paper we use the following abbreviations:
<list list-type="simple">
<list-item>{x} — <italic>fractional part of x</italic>;</list-item>
<list-item>{x} — <italic>x mod</italic> 1;</list-item>
<list-item>[x] — <italic>integer part of x</italic>;</list-item>
<list-item>u.d. — <italic>uniform distribution</italic>;</list-item>
<list-item>d.f. — <italic>distribution function</italic>;</list-item>
<list-item>a.d.f. — <italic>asymptotic distribution function</italic>;</list-item>
<list-item>u.d.p. — <italic>uniform distribution preserving</italic>;</list-item>
<list-item>step d.f. — <italic>step distribution function</italic>;</list-item>
<list-item>a.e. — <italic>almost everywhere</italic>;</list-item>
<list-item>#X — <italic>cardinality of the set X</italic>.</list-item>
</list>
</abstract>

Slovak University of Technology in Bratislava

Copulas

AHEAD OF PRINT

Volume 18 (2023): Issue 1 (July 2023)

Volume 17 (2022): Issue 2 (December 2022)

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

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

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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)

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. 