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Transactions of the American Mathematical Society

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On the dimension spectrum of infinite subsystems of continued fractions


Authors: Vasileios Chousionis, Dmitriy Leykekhman and Mariusz Urbański
Journal: Trans. Amer. Math. Soc. 373 (2020), 1009-1042
MSC (2010): Primary 37D35, 28A80, 11K50, 11J70; Secondary 37B10, 37C30, 37C40
DOI: https://doi.org/10.1090/tran/7984
Published electronically: November 5, 2019
MathSciNet review: 4068257
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Abstract: In this paper we study the dimension spectrum of continued fractions with coefficients restricted to infinite subsets of natural numbers. We prove that if $E$ is any arithmetic progression, the set of primes, or the set of squares $\{n^2\}_{n \in \mathbb {N}}$, then the continued fractions whose digits lie in $E$ have full dimension spectrum, which we denote by $DS(\mathcal {CF}_E)$. Moreover we prove that if $E$ is an infinite set of consecutive powers, then the dimension spectrum $DS(\mathcal {CF}_E)$ always contains a non-trivial interval. We also show that there exists some $E \subset \mathbb {N}$ and two non-trivial intervals $I_1, I_2$, such that $DS(\mathcal {CF}_E) \cap I_1=I_1$ and $DS(\mathcal {CF}_E) \cap I_2$ is a Cantor set. On the way we employ the computational approach of Falk and Nussbaum in order to obtain rigorous effective estimates for the Hausdorff dimension of continued fractions whose entries are restricted to infinite sets.


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Additional Information

Vasileios Chousionis
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3004
MR Author ID: 858546
Email: vasileios.chousionis@uconn.edu

Dmitriy Leykekhman
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3004
MR Author ID: 680657
Email: dmitriy.leykekhman@uconn.edu

Mariusz Urbański
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203-5017
Email: urbanski@unt.edu

Received by editor(s): June 11, 2018
Received by editor(s) in revised form: May 5, 2019
Published electronically: November 5, 2019
Additional Notes: The first author was supported by the Simons Foundation via the project “Analysis and dynamics in Carnot groups”, Collaboration Grant no. 521845
The second author was supported by NSF grant no. DMS-1522555
Article copyright: © Copyright 2019 American Mathematical Society