On distribution of continuous sequences

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

Acta Arithmetica, 1998

HAL (Le Centre pour la Communication Scientifique Directe), 1997

Springer Proceedings in Mathematics & Statistics, 2012

A hybrid sequence in the multidimensional unit cube is a combination of two or more lower-dimensional sequences of different types. In this paper, we present tools to analyze the uniform distribution of such sequences. In particular, we introduce hybrid function systems, which are classes of functions that are composed of the trigonometric functions, the Walsh functions in base b, and the p-adic functions. The latter are related to the dual group of the p-adic integers, p a prime. We prove the Weyl criterion for hybrid function systems and define a new notion of diaphony, the hybrid diaphony. Our approach generalizes several known concepts and results.

Proceedings of the Steklov Institute of Mathematics, 2012

Let f denote an additive arithmetical function with continuous limiting distribution F on the integers. Then f also has a limiting distribution G on shifted primes. Under some growth conditions on the values of f at primes, we provide optimal lower bounds for the modulus of continuity of F and G, at all points from a specified infinite set.

Proceedings of the American Mathematical Society, 2008

We show how different random thin sets of integers may have different behaviour. First, using a recent deviation inequality of Boucheron, Lugosi and Massart, we give a simpler proof of one of our results in Some new thin sets of integers in harmonic analysis, Journal d'Analyse Mathématique 86 (2002), 105-138, namely that there exist 4 3-Rider sets which are sets of uniform convergence and Λ(q)-sets for all q < ∞ but which are not Rosenthal sets. In a second part, we show, using an older result of Kashin and Tzafriri, that, for p > 4 3 , the p-Rider sets which we had constructed in that paper are almost surely not of uniform convergence.

2002

The paper describes ergodic (with respect to the Haar measure) functions in the class of all functions, which are defined on (and take values in) the ring of p-adic integers, and which satisfy (at least, locally) Lipschitz condition with coefficient 1. Equiprobable (in particular, measure-preserving) functions of this class are described also. In some cases (and especially for p=2) the

Acta Arithmetica, 2002

Corrigendum to Theorem 5 of the paper "Asymptotic density of A ⊂ N and density of the ratio set R(A)" (Acta Arith. 87 (1998), 67-78) by Oto Strauch (Bratislava) and János T. Tóth (Ostrava) In the proof of Theorem 5 in [2], step 3 is incorrect. We want to thank S. V. Konyagin who has pointed it out. The wrong Theorem 5 asserts that for every increasing sequence of positive integers x n , n = 1, 2,. .. , with a positive lower asymptotic density, if there exists an interval (u, v) containing no limit points of the ratio sequence x m /x n , m, n = 1, 2,. .. , where u, v are limit points, then there are infinitely many such intervals. In the new form of Theorem 5 we replace intervals (u, v) containing no limit points of x m /x n with intervals having some zero asymptotic density of x m /x n and we reformulate it in terms of distribution functions of x m /x n. We prove that if there exists an interval (u, v), containing no limit points of x m /x n , then every distribution function of x m /x n has infinitely many intervals with constant values, assuming positive lower asymptotic density of x n. For an illustration, we give two examples. In Example 1, x m /x n has only one such interval (u, v), and in Example 2 it has infinitely many, and in both cases every distribution function of x m /x n has infinitely many intervals with constant values. Finally, we discuss via Examples 1 and 2 a possibility of adding a proposition contained in the incorrect step 3 as an assumption of Theorem 5. To do this we need the following concept used in [3] (see [1] for a general account). A function g : [0, 1] → [0, 1] will be called a distribution function (abbreviated d.f.) if g(0) = 0, g(1) = 1, and g is nondecreasing. We will identify any two distribution functions coinciding a.e. on [0, 1]. A point β ∈ [0, 1]

1996

In the first part of the paper we define the notion of the density as certain type of finitely additive probability measure and the distribution function of sequences with respect to the density. Then we derive some simple criterions providing the continuity of the distribution function of given sequence. These criterions we apply to the van der Corput's sequences. The Weyl's type criterions of continuity of the distribution function are proven.