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Oto Strauch

ABSTRACT

In this paper there are given problems from the Unsolved Problems Section on the homepage of the journal Uniform Distribution Theory <http://www.boku.ac.at/MATH/udt/unsolvedproblems.pdf> It contains 38 items and 5 overviews collected by the author and by Editors of UDT. They are focused at uniform distribution theory, more accurate, distri- bution functions of sequences, logarithm of primes, Euler totient function, van der Corput sequence, ratio sequences, set of integers of positive density, exponen- tial sequences, moment problems, Benford’s law, Gauss-Kuzmin theorem, Duffin- Schaeffer conjecture, extremes fQ fQ F(x,y)dg(x,y) over copulas g(x,y), sum- -of-digits sequence, etc. Some of them inspired new research activities. The aim of this printed version is publicity.

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Oto Strauch

Abstract

This expository paper presents known results on distribution functions g(x) of the sequence of blocks where xn is an increasing sequence of positive integers. Also presents results of the set G(Xn) of all distribution functions g(x). Specially:

- continuity of g(x);

- connectivity of G(Xn);

- singleton of G(Xn);

- one-step g(x);

- uniform distribution of Xn, n = 1, 2, . . . ;

- lower and upper bounds of g(x);

- applications to bounds of ;

- many examples, e.g., , where pn is the nth prime, is uniformly distributed.

The present results have been published by 25 papers of several authors between 2001-2013.

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Yukio Ohkubo and Oto Strauch

Abstract

Applying the theory of distribution functions of sequences we find the relative densities of the first digits also for sequences xn not satisfying Benford’s law. Especially for sequence xn = nr, n = 1, 2, . . . and xn=pnr, n = 1, 2, . . ., where pn is the increasing sequence of all primes and r > 0 is an arbitrary real. We also add rate of convergence to such densities.

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Vladimír Baláž, Maria Rita Iacò, Oto Strauch, Stefan Thonhauser and Robert F. Tichy

Abstract

In this paper we consider an optimization problem for Cesàro means of bivariate functions. We apply methods from uniform distribution theory, calculus of variations and ideas from the theory of optimal transport.

Open access

Vladimír Baláž, Jana Fialová, Markus Hofer, Maria R. Iacò and Oto Strauch

Abstract

Let γq(n) be the van der Corput sequence in the base q and g(x, y, z, u) be an asymptotic distribution function of the 4-dimensional sequence

In this paper we find an explicit formula for g(x, x, x, x) and then as an example we find the limit

for the base q = 4, 5, 6, . . . Also we find an explicit form of sth iteration T(s)(x) of the von Neumann-Kakutani transformation defined by T(γq(n)) = γq(n + 1).