Metric Diophantine Approximation—From Continued Fractions to Fractals

Kristensen, Simon

doi:10.1007/978-3-319-48817-2_2

Cited by 4 publications

(4 citation statements)

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“…Metrical theory of continued fractions plays a significant role in the theory of metric Diophantine approximation. We state some useful basic properties of continued fractions of real numbers and recommend the reader to [14,17] for further details.…”

Section: Preliminaries and Auxiliary Resultsmentioning

confidence: 99%

The sets of Dirichlet non-improvable numbers versus well-approximable numbers

Bakhtawar

Bos

Hussain

2019

Ergod. Th. Dynam. Sys.

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Section: Preliminaries and Auxiliary Resultsmentioning

confidence: 99%

The sets of Dirichlet non-improvable numbers versus well-approximable numbers

Bakhtawar

Bos

Hussain

2019

Ergod. Th. Dynam. Sys.

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“…In this section, we gather some fundamental properties of continued fractions of real numbers and recommend [10,14] to the reader for further details.…”

Section: Preliminariesmentioning

confidence: 99%

“…Therefore, (4-17) is true for r = k + 1. Thus, by using (4-16) and (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17), it is easy to check that the following equality holds:…”

Section: This Implies Thatmentioning

confidence: 99%

Hausdorff Dimension for the Set of Points Connected With the Generalized Jarník–besicovitch Set

Bakhtawar

2020

J. Aust. Math. Soc.

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In this article we aim to investigate the Hausdorff dimension of the set of points $x \in [0,1)$ such that for any $r\in \mathbb {N}$ , $$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{\tau(x)(h(x)+\cdots+h(T^{n-1}(x)))} \end{align*} $$ holds for infinitely many $n\in \mathbb {N}$ , where h and $\tau $ are positive continuous functions, T is the Gauss map and $a_{n}(x)$ denotes the nth partial quotient of x in its continued fraction expansion. By appropriate choices of $r,\tau (x)$ and $h(x)$ we obtain various classical results including the famous Jarník–Besicovitch theorem.

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“…Their influential investigations have led to a blossoming area broadly known as metric Diophantine approximation, with several connections to classical number theoretic questions, as well as more surprising links to mathematical physics, dynamical systems, fractal geometry, analytic combinatorics, computer science, wireless communication, etc. -see [6,7,13,23,27,29,33,44,48,57,62,64,71,81,84] and the references therein for a sampling of such relationships.…”

Section: Introductionmentioning

confidence: 99%

Hausdorff dimensions of perturbations of a conformal iterated function system via thermodynamic formalism

Das¹,

Fishman²,

Simmons³

et al. 2020

Preprint

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We consider small perturbations of a conformal iterated function system (CIFS) produced by either adding or removing some generators with small derivative from the original. We establish a formula, utilizing transfer operators arising from the thermodynamic formalism à la Sinai-Ruelle-Bowen, which may be solved to express the Hausdorff dimension of the perturbed limit set in series form: either exactly, or as an asymptotic expansion. Significant applications include strengthening Hensley's asymptotic formula from 1992, which improved on earlier bounds due to Jarník and Kurzweil, for the Hausdorff dimension of the set of real numbers whose continued fraction expansion partial quotients are all ≤ N ; as well as its counterpart for reals whose partial quotients are all ≥ N due to Good from 1941. CONTENTS1. Introduction 2 2. An abstract operator formula 5 3. Point-accumulating conformal iterated function systems 8 4. Statement of main result 14 5. Spectral gap in the case c > 0 17 6. Proof of Theorem 4.1 19 7. Simplifications in special cases 24 8. Solving (4.1): a motivating computation 28 9. Gauss IFS Examples 30 10. Directions to further research 41 Appendix A. Computation of coefficients 42 Appendix B. Definition of a conformal iterated function system (CIFS) 45 References 46

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Metric Diophantine Approximation—From Continued Fractions to Fractals

Abstract: We outline a proof of an analogue of Khintchine's Theorem in R 2 , where the ordinary height is replaced by a distance function satisfying an irrationality condition as well as certain decay and symmetry conditions.

Cited by 4 publications

References 45 publications

The sets of Dirichlet non-improvable numbers versus well-approximable numbers

The sets of Dirichlet non-improvable numbers versus well-approximable numbers

Hausdorff Dimension for the Set of Points Connected With the Generalized Jarník–besicovitch Set

Hausdorff dimensions of perturbations of a conformal iterated function system via thermodynamic formalism

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