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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the dimension spectrum of infinite subsystems of continued fractions
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by Vasileios Chousionis, Dmitriy Leykekhman and Mariusz Urbański PDF
Trans. Amer. Math. Soc. 373 (2020), 1009-1042 Request permission

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
  • © Copyright 2019 American Mathematical Society
  • 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
  • MathSciNet review: 4068257