Про одне узагальнення поняття нормальних чисел

The paper considers the generalization of the concept of normal numbers in the context of the classical s-th representation of real numbers, in relation to the Q_s-representation, first considered by M. Pratsiovytyi. The result of I. Nivena and H. Zukerman is deepened in relation to the metric theory of normal E. Borel numbers. It is shown that the set of all Q_s-normal numbers has a Lebesgue measure 1. The connection between the property of normality and the uniform distribution of the sequence of numbers generated by the shift operator in relation to the corresponding number is established. It was found that the set of all numbers of the segment [0; 1] for which the corresponding sequence generated by the operator of left-hand shift Q_s-digits is uniformly distributed has a full Lebesgue measure. The corresponding theorems deepen the results of the metric theory Q_s-decompositions of real numbers of the segment [0; 1] obtained by M. Pratsiovytyi and G. Torbin. Pages of the article in the issue: 58 - 62 Language of the article: UkrainianВ роботі розглядається узагальнення поняття нормальних чисел, в контексті класичного $s$-го представлення дійсних чисел, по відношенню до $Q_s$-представлення, вперше розглянутого М. Працьовитим. Поглиблюється результат І.~Нівена та Г.~Цукермана, по відношенню до метричної теорії нормальних чисел Е.~Бореля. Показано, що множина всіх $Q_s$-нормальних чисел має міру Лебега $1$. Встановлюється зв'язок між властивістю нормальності та рівномірною розподіленістю послідовності чисел, породжених оператором зсуву по відношенню до відповідного числа. Було встановлено, що множина всіх чисел відрізку $[0;1]$ для яких відповідна послідовність породжена оператором лівостороннього зсуву $Q_s$-цифр є рівномірно розподіленою, має повну міру Лебега. Відповідні теореми поглиблюють результати метричної теорії $Q_s$-розкладів дійсних чисел відрізку $[0;1]$ отриманих М.~Працьовитим та Г.~Торбіним

Old and new results on normality

We present a partial survey on normal numbers, including Keane's contributions, and with recent developments in different directions.Comment: Published at http://dx.doi.org/10.1214/074921706000000248 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

A Weyl Criterion for Finite-State Dimension and Applications

Finite-state dimension, introduced early in this century as a finite-state version of classical Hausdorff dimension, is a quantitative measure of the lower asymptotic density of information in an infinite sequence over a finite alphabet, as perceived by finite automata. Finite-state dimension is a robust concept that now has equivalent formulations in terms of finite-state gambling, lossless finite-state data compression, finite-state prediction, entropy rates, and automatic Kolmogorov complexity. The Schnorr-Stimm dichotomy theorem gave the first automata-theoretic characterization of normal sequences, which had been studied in analytic number theory since Borel defined them. This theorem implies that a sequence (or a real number having this sequence as its base-b expansion) is normal if and only if it has finite-state dimension 1. One of the most powerful classical tools for investigating normal numbers is the Weyl criterion, which characterizes normality in terms of exponential sums. Such sums are well studied objects with many connections to other aspects of analytic number theory, and this has made use of Weyl criterion especially fruitful. This raises the question whether Weyl criterion can be generalized from finite-state dimension 1 to arbitrary finite-state dimensions, thereby making it a quantitative tool for studying data compression, prediction, etc. This paper does exactly this. We extend the Weyl criterion from a characterization of sequences with finite-state dimension 1 to a criterion that characterizes every finite-state dimension. This turns out not to be a routine generalization of the original Weyl criterion. Even though exponential sums may diverge for non-normal numbers, finite-state dimension can be characterized in terms of the dimensions of the subsequence limits of the exponential sums. We demonstrate the utility of our criterion though examples