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Athanasios Sourmelidis

Abstract

In this paper, we prove a discrete analogue of Voronin’s early finite-dimensional approximation result with respect to terms from a given Beatty sequence and make use of Taylor approximation in order to derive a weak universality statement.

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Martine Queffélec

Abstract

We intend to unroll the surprizing properties of the Thue-Morse sequence with a harmonic analysis point of view, and mention in passing some related open questions.

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Florian Pausinger and Alev Topuzoğlu

Abstract

A permuted van der Corput sequence Sbσ in base b is a one-dimensional, infinite sequence of real numbers in the interval [0, 1), generation of which involves a permutation σ of the set {0, 1,..., b − 1}. These sequences are known to have low discrepancy DN, i.e. t(Sbσ):=limsupNDN(Sbσ)/logN is finite. Restricting to prime bases p we present two families of generating permutations. We describe their elements as polynomials over finite fields 𝔽p in an explicit way. We use this characterization to obtain bounds for t(Spσ) for permutations σ in these families. We determine the best permutations in our first family and show that all permutations of the second family improve the distribution behavior of classical van der Corput sequences in the sense that t(Spσ)<t(Spid).

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Anshika Srivastava, Ram Krishna Pandey and Om Prakash

Abstract

This paper concerns the problem of determining or estimating the maximal upper density of the sets of nonnegative integers S whose elements do not differ by an element of a given set M of positive integers. We find some exact values and some bounds for the maximal density when the elements of M are generalized Fibonacci numbers of odd order. The generalized Fibonacci sequence of order r is a generalization of the well known Fibonacci sequence, where instead of starting with two predetermined terms, we start with r predetermined terms and each term afterwards is the sum of r preceding terms. We also derive some new properties of the generalized Fibonacci sequence of order r. Furthermore, we discuss some related coloring parameters of distance graphs generated by the set M.

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Mario Neumüller and Friedrich Pillichshammer

Abstract

The star discrepancy DN*(𝒫) is a quantitative measure for the irregularity of distribution of a finite point set 𝒫 in the multi-dimensional unit cube which is intimately related to the integration error of quasi-Monte Carlo algorithms. It is known that for every integer N ≥ 2 there are point sets 𝒫 in [0, 1)d with |𝒫| = N and DN*(𝒫)=O((logN)d-1/N). However, for small N compared to the dimension d this asymptotically excellent bound is useless (e.g., for N ≤ ed −1).

In 2001 it has been shown by Heinrich, Novak, Wasilkowski and Woźniakowski that for every integer N ≥ 2there exist point sets 𝒫 in [0, 1)d with |𝒫| = N and DN*(𝒫)Cd/N. Although not optimal in an asymptotic sense in N, this upper bound has a much better (and even optimal) dependence on the dimension d.

Unfortunately the result by Heinrich et al. and also later variants thereof by other authors are pure existence results and until now no explicit construction of point sets with the above properties is known. Quite recently Löbbe studied lacunary subsequences of Kronecker’s (n α)-sequence and showed a metrical discrepancy bound of the form Cd(logd)/N with implied absolute constant C> 0 independent of N and d.

In this paper we show a corresponding result for digital Kronecker sequences, which are a non-archimedean analog of classical Kronecker sequences.

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Roswitha Hofer and Ísabel Pirsic

Abstract

We introduce a hybridization of digital sequences with uniformly distributed sequences in the domain of b-adic integers, ℤb,b ∈ℕ \ {1}, by using such sequences as input for generating matrices. The generating matrices are then naturally required to have finite row-lengths. We exhibit some relations of the ‘classical’ digital method to our extended version, and also give several examples of new constructions with their respective quality assessments in terms of t, T and discrepancy.

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Tamás Herendi

Abstract

The aim of the present paper is to provide the background to construct linear recurring sequences with uniform distribution modulo 2s. The theory is developed and an algorithm based on the achieved results is given. The constructed sequences may have arbitrary large period length depending only on the computational power of the used machines.

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Adam Grabowski

Summary

In the article we formalize some properties needed to prove that sequences of prime reciprocals are divergent. The aim is to show that the series exhibits log-log growth. We introduce some auxiliary notions as harmonic numbers, telescoping series, and prove some standard properties of logarithms and exponents absent in the Mizar Mathematical Library. At the end we proceed with square-free and square-containing parts of a natural number and reciprocals of corresponding products.