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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Normal numbers with digit dependencies


Authors: Christoph Aistleitner, Verónica Becher and Olivier Carton
Journal: Trans. Amer. Math. Soc. 372 (2019), 4425-4446
MSC (2010): Primary 11K16; Secondary 68R15
DOI: https://doi.org/10.1090/tran/7706
Published electronically: November 26, 2018
MathSciNet review: 4009433
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Abstract: We give metric theorems for the property of Borel normality for real numbers under the assumption of digit dependencies in their expansion in a given integer base. We quantify precisely how much digit dependence can be allowed such that almost all real numbers are normal. Our theorem states that almost all real numbers are normal when at least slightly more than $ \log \log n$ consecutive digits with indices starting at position $ n$ are independent. As the main application, we consider the Toeplitz set $ T_P$, which is the set of all sequences $ a_1a_2 \ldots $ of symbols from $ \{0, \ldots , b-1\}$ such that $ a_n$ is equal to $ a_{pn}$ for every $ p$ in $ P$ and $ n=1,2,\ldots $. Here $ b$ is an integer base and $ P$ is a finite set of prime numbers. We show that almost every real number whose base $ b$ expansion is in $ T_P$ is normal to base $ b$. In the case when $ P$ is the singleton set $ \{2\}$ we prove that more is true: almost every real number whose base $ b$ expansion is in $ T_P$ is normal to all integer bases. We also consider the Toeplitz transform which maps the set of all sequences to the set $ T_P$, and we characterize the normal sequences whose Toeplitz transform is normal as well.


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Additional Information

Christoph Aistleitner
Affiliation: Institute of Analysis and Number Theory, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
Email: aistleitner@math.tugraz.at

Verónica Becher
Affiliation: Departamento de Computación and ICC CONICET, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, C1428EGA Buenos Aires, Argentina
Email: vbecher@dc.uba.ar

Olivier Carton
Affiliation: Institut de Recherche en Informatique Fondamentale, Université Paris Diderot, Case 7014, 75205 Paris Cedex 13, France
Email: olivier.carton@irif.fr

DOI: https://doi.org/10.1090/tran/7706
Received by editor(s): April 9, 2018
Received by editor(s) in revised form: June 26, 2018, and September 4, 2018
Published electronically: November 26, 2018
Additional Notes: The first author is supported by the Austrian Science Fund (FWF), projects Y-901 and F-5512-N26.
The second and third authors are members of the Laboratoire International Associé INFINIS, CONICET/Universidad de Buenos Aires–CNRS/Université Paris Diderot, and they are supported by the ECOS project PA17C04.
The third author is also partially funded by the DeLTA project (ANR-16-CE40-0007).
Article copyright: © Copyright 2018 American Mathematical Society