Pollicott, Mark and Vytnova, Polina (2022) Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups. Transactions of the American Mathematical Society, 9 . pp. 1102-1159. doi:10.1090/btran/109 ISSN 0002-9947.

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Abstract

In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable attention, but we are particularly concerned with the role of the value of the Hausdorff dimension in solving conjectures and problems in other areas of mathematics. As our first application we confirm, and often strengthen, conjectures on the difference of the Lagrange and Markov spectra in Diophantine analysis, which appear in the work of Matheus and Moreira [Comment. Math. Helv. 95 (2020), pp. 593–633]. As a second application we (re-)validate and improve estimates connected with the Zaremba conjecture in number theory, used in the work of Bourgain–Kontorovich [Ann. of Math. (2) 180 (2014), pp. 137–196], Huang [An improvement to Zaremba’s conjecture, ProQuest LLC, Ann Arbor, MI, 2015] and Kan [Mat. Sb. 210 (2019), pp. 75–130]. As a third more geometric application, we rigorously bound the bottom of the spectrum of the Laplacian for infinite area surfaces, as illustrated by an example studied by McMullen [Amer. J. Math. 120 (1998), pp. 691-721].

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Hausdorff measures , Lagrange equations , Markov spectrum , Fuchsian groups , Diophantine analysis , Number theory
Journal or Publication Title:	Transactions of the American Mathematical Society
Publisher:	American Mathematical Society
ISSN:	0002-9947
Official Date:	30 December 2022
Dates:	Date Event 30 December 2022 Published 17 January 2022 Accepted
Volume:	9
Page Range:	pp. 1102-1159
DOI:	10.1090/btran/109
Status:	Peer Reviewed
Publication Status:	Published
Reuse Statement (publisher, data, author rights):	First published in Transactions of the American Mathematical Society in [volume/issue number and year], published by the American Mathematical Society.
Access rights to Published version:	Open Access (Creative Commons)
Copyright Holders:	© Copyright 2022 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
Date of first compliant deposit:	17 January 2022
Date of first compliant Open Access:	21 January 2022
RIOXX Funder/Project Grant:	Project/Grant ID RIOXX Funder Name Funder ID 833802 (Resonances) European Research Council http://dx.doi.org/10.13039/501100000781 EP/T001674/1 [EPSRC] Engineering and Physical Sciences Research Council http://dx.doi.org/10.13039/501100000266 EP/T001674/1 [EPSRC] Engineering and Physical Sciences Research Council http://dx.doi.org/10.13039/501100000266
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