Volume 56 (2013): Issue 1 (November 2013)
Number Theory

The paper contains a short prehistory and history of the Czech and Slovak international number theory conferences

We study algebraic properties of real positive algebraic numbers which are not less than the moduli of their conjugates. In particular, we are interested in the relation of these numbers to Perron numbers.

In this paper we study the conditions (1), (2) and (3) for the permutations which preserve the weighted density. These conditions are motivated by the conditions of Lévy group, originated in [Levy, P.: Problèmes concrets d’Analyse Fonctionelle. Gauthier Villars, Paris, 1951], and studied in [Obata, N.: Density of natural numbers and Lévy group, J. Number Theory 30 (1988), 288-297]. In the second part we prove that under some conditions for the sequence of weights, for instance for the logarithmic density, the first two conditions can be launched

The sets of the form C + mC, where m ℮ (0, 1) and C is the classic Cantor ternary set, are described.

We define uniform distribution in compact metric space with respect to the Buck’s measure density originated in [Buck, R. C.: The measure theoretic approach to density, Amer. J. Math. 68 (1946), 560-580]. Weyl’s criterion is derived. This leads to an existence result.

The concept of Wijsman statistical convergence was defined by [Nuray, F.-Rhoades, B. E.: Statistical convergence of sequences of sets, Fasc. Math. 49 (2012), 1-9]. In this paper we present three definitions which are a natural combination of the definition of asymptotic equivalence, statistical convergence, generalized statistical convergence and Wijsman convergence. In addition, we also present asymptotically equivalent sequences of sets in sense of Wijsman and study some properties of this concept.

An analogue of the convergence part of Khintchine’s theorem (1924) for simultaneous approximation of integral polynomials at the points (x1, x2, z,w) ∈ R2 × C × Qp is proved. It is a solution of the more general problem than Sprindźuk's problem (1980) in the ring of adeles. We use a new form of the essential and nonessential domain methods in metric theory of Diophantine approximation

Let K/Q be a cyclic cubic field with an prime conductor l. In the paper there is given method for verification that K is ω-good and it is applied for conductors up to l = 349.

The skew-harmonic numbers are the partial sums of the alternating harmonic series, i.e., the expansion of log 2.We evaluate in closed form various power series and numerical series with skew-harmonic numbers. This provides a simultaneous solution of two recent problems by Ovidiu Furdui in the American Mathematical Monthly and the College Mathematics Journal. We also present and discuss representations involving the dilogarithm and the trilogarithm which are related to our results. Finally, we provide the evaluations of several double integrals in terms of classical constants.

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 f_Q f_Q 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.