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

#### Search

#### Dedicated to the fifth international conference on Uniform Distribution Theory (UDT 2016) Sopron, Hungary, July 5–8, 2016

- Open Access

The Fifth International Conference on Uniform Distribution Theory (UDT 2016) Sopron, Hungary, July 5–8, 2016

Page range: i - vii

#### Abstract

This volume contains papers originally presented or inspired by the Fifth International Conference on Uniform Distribution Theory which was held in Sopron, Hungary, July 5-8, 2016.

- Open Access

The Bb-adic Symmetrization of Digital Nets for Quasi-Monte Carlo Integration

Page range: 1 - 25

#### Abstract

The notion of symmetrization, also known as Davenport’s reflection principle, is well known in the area of the discrepancy theory and quasi- Monte Carlo (QMC) integration. In this paper we consider applying a symmetrization technique to a certain class of QMC point sets called digital nets over ℤ_{b}. Although symmetrization has been recognized as a geometric technique in the multi-dimensional unit cube, we give another look at symmetrization as a geometric technique in a compact totally disconnected abelian group with dyadic arithmetic operations. Based on this observation we generalize the notion of symmetrization from base 2 to an arbitrary base b ∈ ℕ, b ≥ 2. Subsequently, we study the QMC integration error of symmetrized digital nets over ℤ_{b} in a reproducing kernel Hilbert space. The result can be applied to component-by-component construction or Korobov construction for finding good symmetrized (higher order) polynomial lattice rules which achieve high order convergence of the integration error for smooth integrands at the expense of an exponential growth of the number of points with the dimension. Moreover, we consider two-dimensional symmetrized Hammersley point sets in prime base b, and prove that the minimum Dick weight is large enough to achieve the best possible order of L_{p} discrepancy for all 1 ≤ p < ∞.

#### Keywords

- Quasi-Monte Carlo
- b-adic symmetrization
- digital nets
- Hammersley point sets

- Open Access

Individual Gap Measures from Generalized Zeckendorf Degompositions

Page range: 27 - 36

#### Abstract

Zeckendorf's theorem states that every positive integer can be decomposed uniquely as a sum of nonconsecutive Fibonacci numbers. The distribution of the number of summands converges to a Gaussian, and the individual measures on gajw between summands for m € [F_{n},F_{n+1}) converge to geometric decay for almost all m as n→ ∞. While similar results are known for many other recurrences, previous work focused on proving Gaussianity for the number of summands or the average gap measure. We derive general conditions, which are easily checked, that yield geometric decay in the individual gap measures of generalized Zerkendorf decompositions attached to many linear recurrence relations.

#### Keywords

- Zeckendorf decompositions
- individual gap measures
- Levy’s Criterion

- Open Access

p-adic Valuation of Exponential Sums in One Variable Associated to Binomials

Page range: 37 - 53

#### Abstract

In this paper we compute the p-adic valuation of exponential sums associated to binomials F(X) = aX^{d₁} + bX^{d₂} over F_{p}. In particular, its p-adic valuation is constant for a, b ∈ F∗_{p} . As a byproduct of our results, we obtain a lower bound for the sizes of value sets of binomials over F_{q}.

#### Keywords

- p-divisibility
- exponential sums
- value sets

- Open Access

On Irregularities of Distribution of Binary Sequences Relative to Arithmetic Progressions, I. (General Results)

Page range: 55 - 67

#### Abstract

In 1964 K. F. Roth initiated the study of irregularities of distribution of binary sequences relative to arithmetic progressions and since that numerous papers have been written on this subject. In the applications one needs binary sequences which are well distributed relative to arithmetic progressions, in particular, in cryptography one needs binary sequences whose short subsequences are also well-distributed relative to arithmetic progressions. Thus we introduce weighted measures of pseudorandomness of binary sequences to study this property. We study the typical and minimal values of this measure for binary sequences of a given length.

#### Keywords

- arithmetic progressions
- well-distribution
- binary sequence

- Open Access

Integral Powers of Numbers in Small Intervals Modulo 1: The Cardinality Gap Phenomenon

Page range: 69 - 98

#### Abstract

This paper deals with the distribution of αζ^{n} mod 1, where α ≠ 0, ζ > 1 are fixed real numbers and n runs through the positive integers. Denote by ‖·‖ the distance to the nearest integer. We investigate the case of αζ^{n} all lying in prescribed small intervals modulo 1 for all large n, with focus on the case ‖αζ^{n}‖ ≤ ɛ for small ɛ > 0. We are particularly interested in what we call cardinality gap phenomena. For example for fixed ζ > 1 and small ɛ > 0 there are at most countably many values of α such that ‖αζ^{n}‖ ≤ ɛ for all large n, whereas larger ɛ induces an uncountable set. We investigate the value of ‖ at which the gap occurs. We will pay particular attention to the case of algebraic and, more specific, rational ζ > 1. Results concerning Pisot and Salem numbers such as some contribution to Mahler’s 3/2-problem are implicitly deduced. We study similar questions for fixed α ≠ 0 as well.

#### Keywords

- distribution modulo 1
- distribution of powers
- Pisot numbers
- Salem numbers
- Mahler’s 3/2-problem

- Open Access

Additive Energy and Irregularities of Distribution

Page range: 99 - 107

#### Abstract

We consider strictly increasing sequences (a_{n})_{n≥1} of integers and sequences of fractional parts ({a_{n}α})_{n≥1} where α ∈ R. We show that a small additive energy of (a_{n})_{n≥1} implies that for almost all α the sequence ({a_{n}α})_{n≥1} has large discrepancy. We prove a general result, provide various examples, and show that the converse assertion is not necessarily true.

#### Keywords

- discrepancy
- additive energy
- metric number theory
- additive combinatorics

- Open Access

Statistical Distribution of Roots of a Polynomial Modulo Primes II

Page range: 109 - 122

#### Abstract

Continuing the previous paper, we give several data on the distribution of roots modulo primes of an irreducible polynomial, and based on them, we propose problems on the distribution.

#### Keywords

- distribution
- polynomial

- Open Access

Uniform Distribution with Respect to Density

Page range: 123 - 138

#### Abstract

The paper deals with a generalisation of uniform distribution. The analogues of Weyl’s criterion are derived.

#### Keywords

- uniform distribution
- density
- Riemann integration

- Open Access

Une Propriété Topologique de Certains Ensembles de Mills

Page range: 139 - 153

#### Abstract

In this article , we show that the set of Mills constants (real numbers M such that [M^{3ⁿ}] is prime for all n ≥ 0) is the increasing limit of sets homeomorphic to the triadic Cantor’s set. More generally, for a given function ϕ and a set A of integers, we studying the Mills set M_{ϕ}(A) = {α ∈ ℝ/ ∀n ∈ ℕ, [ϕ_{n}(α)] ∈ A} (where ϕ_{n} = ϕ∘...∘ϕ n times). We show that, under certain assumptions over ϕ and A, for all real w > infM_{ϕ}(A) the set M_{ϕ}(A) ∩ [2, w] is homeomorphic to the triadic Cantor’s set.

#### Keywords

- Nombres premiers
- théorème de Mills
- ensemble triadique de Cantor