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ZAREMBA, SALEM AND THE FRACTAL NATURE OF GHOST DISTRIBUTIONS

Part of: Classical measure theory Sequences and sets

Published online by Cambridge University Press:  06 October 2022

MICHAEL COONS ,
JAMES EVANS ,
ZACHARY GROTH  and
NEIL MAÑIBO
MICHAEL COONS*
Affiliation:
Department of Mathematics and Statistics, California State University, Chico, CA 95929, USA
JAMES EVANS
Affiliation:
School of Information and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia e-mail: [email protected]
ZACHARY GROTH
Affiliation:
School of Information and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia e-mail: [email protected]
NEIL MAÑIBO
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

Motivated by near-identical graphs of two increasing continuous functions—one related to Zaremba’s conjecture and the other due to Salem—we provide an explicit connection between fractals and regular sequences by showing that the graphs of ghost distributions, the distribution functions of measures associated to regular sequences, are sections of self-affine sets. Additionally, we provide a sufficient condition for such measures to be purely singular continuous. As a corollary, and analogous to Salem’s strictly increasing singular continuous function, we show that the ghost distributions of the Zaremba sequences are singular continuous.

Keywords

automatic sequenceregular sequenceaperiodic ordersymbolic dynamicsdilation equationself-affine setcontinuous measure

MSC classification

Primary: 11B85: Automata sequences
Secondary: 28A80: Fractals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 3 , June 2023 , pp. 374 - 389
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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