On absolutely normal numbers and their discrepancy estimate

Bulletin of the Australian Mathematical Society, 2011

Given an integer d≥2, a d-normal number, or simply a normal number, is an irrational number whosed-ary expansion is such that any preassigned sequence, of length k≥1, taken within this expansion occurs at the expected limiting frequency, namely 1/dk. Answering questions raised by Igor Shparlinski, we show that 0.P(2)P(3)P(4)…P(n)… and 0.P(2+1)P(3+1)P(5+1)…P(p+1)…, where P(n) stands for the largest prime factor of n, are both normal numbers.

2010

A real number is called normal if every block of digits in its expansion occurs with the same frequency. A famous result of Borel is that almost every number is normal. Our paper presents an elementary proof of that fact using properties of a special class of functions.

Let (x(n)) n≥1 be an s-dimensional Niederreiter-Xing's sequence in base b. Let D((x(n)) N n=1 ) be the discrepancy of the sequence In this paper, we prove that this estimate is exact. Namely, there exists a constant K > 0, such that inf We also get similar results for other explicit constructions of (t, s) sequences.

arXiv (Cornell University), 2023

Given an integer b 2 and a set P of prime numbers, the set T P of Toeplitz numbers comprises all elements of [0, b[ whose digits (an) n 1 in the base-b expansion satisfy an = apn for all p ∈ P and n 1. Using a completely additive arithmetical function, we construct a number in T P that is simply Borel normal if, and only if, p ∈P 1/p = ∞. We then provide an effective bound for the discrepancy.

Cornell University - arXiv, 2022

The main interest of this article is the one-sided boundedness of the local discrepancy of α ∈ R \ Q on the interval (0, c) ⊂ (0, 1) defined by Dn(α, c) = n j=1 1 {{jα}<c} − cn. We focus on the special case c ∈ (0, 1) ∩ Q. Several necessary and sufficient conditions on α for (Dn(α, c)) to be one-side bounded are derived. Using these, certain topological properties are given to describe the size of the set Oc = {α ∈ R + \ Q : (Dn(α, c)) is one-side bounded}.

1990

Experimental Mathematics, 2001

Bailey and Crandall [4] recently formulated "Hypothesis A", a general principle to explain the (conjectured) normality of the binary expansion of constants like π and other related numbers, or more generally the base b expansion of such constants for an integer b ≥ 2. This paper shows that a basic mechanism underlying their principle, which is a relation between single orbits of two discrete dynamical systems, holds for a very general class of representations of numbers. This general class includes numbers for which the conclusion of "Hypothesis A" is not true. The paper also relates the particular class of arithmetical constants treated by Bailey and Crandall to special values of G-functions, and points out an analogy of "Hypothesis A" with Furstenberg's conjecture on invariant measures.

Journal of Number Theory, 1981

Acta Arithmetica, 2009

Dedicated to Harald Niederreiter on the occasion of his 65th birthday 1. Introduction. The inequality of Erdős-Turán-Koksma gives an upper bound for the discrepancy of a finite sequence ω in [0, 1[ s in terms of certain exponential sums; see the monographs Drmota and Tichy [1] and Kuipers and Niederreiter [7] for its general form, and Niederreiter [10] for versions adapted to certain sequences ω of rationals as they appear in applications. These inequalities are an important tool to assess the uniform distribution of low-discrepancy point sets or correlation properties of pseudo-random numbers; see the surveys of Niederreiter [11], Niederreiter and Shparlinski [13] and Hellekalek [4]. The classical inequality of Erdős-Turán-Koksma is based on the trigonometric function system. Variants have been established for Walsh and Haar function systems in an arbitrary integer base b ≥ 2 in Hellekalek [2, 3]. What is the importance of these variants? Different types of sequences ω require different types of exponential sums to study their equidistribution properties, by means of discrepancy and other figures of merit. Hence, by varying the function system, one is able to "synchronize" the exponential sums with the type of sequence under study. In this paper, we will introduce a function system closely related to the dual group of p-adic integers Z p , p a prime, and we will prove a new variant of the inequality of Erdős-Turán-Koksma. This leads to general upper bounds for discrepancy (see Theorem 3.6 and Corollary 3.7). In addition, we will prove a variant of the Weyl criterion for the p-adic function system under consideration (see Theorem 3.8). The uniform distribution of the van der Corput sequence in base p then follows as a simple consequence (see Corollary 3.9).

Annales mathématiques du Québec, 2017

We show that some sequences of real numbers involving sharp normal numbers or non-Liouville numbers are uniformly distributed modulo 1. In particular, we prove that if τ (n) stands for the number of divisors of n and α is a binary sharp normal number, then the sequence (ατ (n)) n≥1 is uniformly distributed modulo 1 and that if g(x) is a polynomial of positive degree with real coefficients and whose leading coefficient is a non-Liouville number, then the sequence (g(τ (τ (n)))) n≥1 is also uniformly distributed modulo 1. Résumé Nous montrons que certaines suites de nombres réels impliquant des nombres normaux robustes et des nombres non-Liouville sont uniformément réparties modulo 1. En particulier, nous démontrons que si τ (n) représente le nombre de diviseurs de n, alors,étant donné un nombre normal binaire robuste α, la suite correspondante (ατ (n)) n≥1 est uniformément répartie modulo 1 et nous démontronségalement que si g(x) est un polynômeà coefficients réels de degré positif et dont le coefficient principal est un nombre non-Liouville, alors la suite (g(τ (τ (n)))) n≥1 est uniformément répartie modulo 1.