A strengthening of a theorem of Bourgain and Kontorovich

Kan, Igor' Davidovich

doi:10.1070/im2014v078n02abeh002689

Cited by 13 publications

(9 citation statements)

References 12 publications

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“…There have been other important refinements on this result by Frolenkov-Kan [12], [13], Kan [25], [26], Huang [18] and Maggee-Oh-Winter [30].…”

Section: Theorem 16 (Bourgain-kontorovich Huang)mentioning

confidence: 78%

“…In the introduction we described interesting results of Bourgain-Kontorovich [2], Huang [18], and Kan [25], [26], [27], which made progress towards the Zaremba Conjecture. These results have a slightly more general formulation, which we will now recall.…”

Section: Zaremba Theorymentioning

confidence: 99%

“…A crucial ingredient in the analysis in the proof of Theorem 1.7 used in [26] is that the limit set E 4 for the iterated function scheme…”

Section: Dimension Estimates For Ementioning

confidence: 99%

“…Theorem 1.7 (Kan [26]). For the alphabet {1, 2, 3, 4} there is a positive density version of the Zaremba conjecture, i.e., lim inf…”

Section: Introductionmentioning

confidence: 99%

“…. In [26] this inequality is attributed to Jenkinson [21], where this value was, in fact, only heuristically estimated. In [23] there is an explicit rigorous bound on the Hausdorff dimension of this set which confirms this inequality.…”

Section: Introductionmentioning

confidence: 99%

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Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups

Pollicott¹,

Vytnova²

2020

Preprint

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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 red of mathematics. As our first application we confirm, and often strengthen, conjectures on the difference of the Lagrange and Markov 1 spectra in Diophantine analysis, which appear in the work of Matheus and Moreira [31]. 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 [2], Huang [18] and Kan [27]. 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 [36].In all approaches to estimating the dimension of limit sets there are questions about the efficiency of the algorithm, the computational effort required and the rigour of the bounds. The approach we use has the virtues of being simple and efficient and we present it in section 3 in a way that is straightforward to implement. * The first author is partly supported by ERC-Advanced Grant 833802-Resonances and EPSRC grant EP/T001674/1 the second author is partly supported by EPSRC grant EP/T001674/1.1 Markov's name will appear in this article in two contexts, namely those of Markov spectra in number theory and the Markov condition from probability theory. Since the both notions are associated with the same person (A.A. Markov, 1856(A.A. Markov, -1922, we have chosen to use the same spelling, despite the conventions often used in these different areas.

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“…There have been other important refinements on this result by Frolenkov-Kan [12], [13], Kan [25], [26], Huang [18] and Maggee-Oh-Winter [30].…”

Section: Theorem 16 (Bourgain-kontorovich Huang)mentioning

confidence: 78%

Section: Zaremba Theorymentioning

confidence: 99%

“…A crucial ingredient in the analysis in the proof of Theorem 1.7 used in [26] is that the limit set E 4 for the iterated function scheme…”

Section: Dimension Estimates For Ementioning

confidence: 99%

“…Theorem 1.7 (Kan [26]). For the alphabet {1, 2, 3, 4} there is a positive density version of the Zaremba conjecture, i.e., lim inf…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups

Pollicott¹,

Vytnova²

2020

Preprint

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A strengthening of a theorem of Bourgain and Kontorovich. III

Kan¹

2015

Izv. Math.

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Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups

Pollicott

Vytnova

2022

Trans. Amer. Math. Soc. Ser. B

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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]. In all approaches to estimating the dimension of limit sets there are questions about the efficiency of the algorithm, the computational effort required and the rigour of the bounds. The approach we use has the virtues of being simple and efficient and we present it in this paper in a way that is straightforward to implement. These estimates apparently cannot be obtained by other known methods.

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A strengthening of a theorem of Bourgain and Kontorovich

Cited by 13 publications

References 12 publications

Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups

Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups

A strengthening of a theorem of Bourgain and Kontorovich. III

Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups

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