<abstract xmlns="http://www.w3.org/1999/xhtml">

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

A Typical Number is Extremely Non-Normal

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"A Typical Number is Extremely Non-Normal"}

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...

A Typical Number is Extremely Non-Normal

Published Online: May 31, 2022

Page range: 77 - 88

Received: Jan 20, 2021

Accepted: Dec 23, 2021

DOI: https://doi.org/10.2478/udt-2022-0001

Keywords
normal numbers, digit frequencies, regular linear transformations

© 2022 Anastasios Stylianou, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

A Typical Number is Extremely Non-Normal

Published Online: May 31, 2022

Page range: 77 - 88

Received: Jan 20, 2021

Accepted: Dec 23, 2021

DOI: https://doi.org/10.2478/udt-2022-0001

Keywordsnormal numbers, digit frequencies, regular linear transformations

© 2022 Anastasios Stylianou, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
normal numbers, digit frequencies, regular linear transformations