On a generalization of the concept of normal numbers

Abstract

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.