On the Behavior of the Spectrum of the Limit Frequencies of Digits under Perturbations of a Real Number

L. N. Pushkin

Kazan State University

Abstract: It is shown that, under the action of random digitwise decaying perturbations not changing the spectrum of the limit frequencies of $r$-adic digits, an arbitrary real number written in the $r$-adic number system becomes almost surely normal over all bases multiplicatively independent of $r$.

Keywords: $r$-adic number system, normal number, digitwise perturbations of a number, spectrum of limit frequencies of digits, Lebesgue measure.

UDC: 511.37+519.21

Received: 09.03.2005

DOI: 10.4213/mzm4150

 English version: Mathematical Notes, 2009, 86:6, 824–830

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