1. bookVolumen 17 (2022): Heft 1 (December 2022)
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Zeitschrift
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2309-5377
Erstveröffentlichung
30 Dec 2013
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A Typical Number is Extremely Non-Normal

Online veröffentlicht: 31 May 2022
Volumen & Heft: Volumen 17 (2022) - Heft 1 (December 2022)
Seitenbereich: 77 - 88
Eingereicht: 20 Jan 2021
Akzeptiert: 23 Dec 2021
Zeitschriftendaten
License
Format
Zeitschrift
eISSN
2309-5377
Erstveröffentlichung
30 Dec 2013
Erscheinungsweise
2 Hefte pro Jahr
Sprachen
Englisch
Abstract

Fix a positive integer N ≥ 2. For a real number x ∈ [0, 1] and a digit i ∈ {0, 1,..., N − 1}, let Πi(x, n) denote the frequency of the digit i among the first nN-adic digits of x. It is well-known that for a typical (in the sense of Baire) x ∈ [0, 1], the sequence of digit frequencies diverges as n →∞. In this paper we show that for any regular linear transformation T there exists a residual set of points x ∈ [0,1] such that the T -averaged version of the sequence (Πi(x, n))n also diverges significantly.

MSC 2010

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