Hausdorff dimension of sets arising in number theory

Exceptional sets related to the largest digits in Lüroth expansions

International Journal of Number Theory ◽

10.1142/s1793042122500737 ◽

2021 ◽

pp. 1-15

Author(s):

Shuyi Lin ◽

Jinjun Li ◽

Manli Lou

Keyword(s):

Number Theory ◽

Hausdorff Dimension ◽

Level Sets ◽

Disjoint Union ◽

Exceptional Set ◽

Exceptional Sets ◽

Lüroth Expansion

Let [Formula: see text] denote the largest digit of the first [Formula: see text] terms in the Lüroth expansion of [Formula: see text]. Shen, Yu and Zhou, A note on the largest digits in Luroth expansion, Int. J. Number Theory 10 (2014) 1015–1023 considered the level sets [Formula: see text] and proved that each [Formula: see text] has full Hausdorff dimension. In this paper, we investigate the Hausdorff dimension of the following refined exceptional set: [Formula: see text] and show that [Formula: see text] has full Hausdorff dimension for each pair [Formula: see text] with [Formula: see text]. Combining the two results, [Formula: see text] can be decomposed into the disjoint union of uncountably many sets with full Hausdorff dimension.

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HAUSDORFF DIMENSION OF THE EXCEPTIONAL SETS CONCERNING THE RUN-LENGTH FUNCTION OF BETA-EXPANSION

Fractals ◽

10.1142/s0218348x18500743 ◽

2018 ◽

Vol 26 (05) ◽

pp. 1850074 ◽

Cited By ~ 1

Author(s):

MENGJIE ZHANG

Keyword(s):

Number Theory ◽

Hausdorff Dimension ◽

Real Number ◽

Maximal Length ◽

Run Length ◽

Exceptional Sets ◽

Number Formula ◽

Consecutive Zero ◽

Almost All ◽

Increasing Function

For any real number [Formula: see text], and any [Formula: see text], let [Formula: see text] be the maximal length of consecutive zeros in the first [Formula: see text] digits of the [Formula: see text]-expansion of [Formula: see text]. Recently, Tong, Yu and Zhao [On the length of consecutive zero digits of [Formula: see text]-expansions, Int. J. Number Theory 12 (2016) 625–633] proved that for any [Formula: see text], for Lebesgue almost all [Formula: see text], [Formula: see text] In this paper, we quantify the size of the set of [Formula: see text] for which [Formula: see text] grows to infinity in a general speed. More precisely, for any increasing function [Formula: see text] with [Formula: see text] tending to [Formula: see text] and [Formula: see text], we show that for any [Formula: see text], the set [Formula: see text] has full Hausdorff dimension.

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Ball-avoiding theorems

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700001000 ◽

2000 ◽

Vol 20 (6) ◽

pp. 1821-1849 ◽

Cited By ~ 2

Author(s):

DOMOKOS SZÁSZ

Keyword(s):

Dynamical System ◽

Number Theory ◽

Phase Space ◽

Hausdorff Dimension ◽

Open Subset ◽

Small Subset ◽

Interested Reader ◽

Rigorous Results ◽

Hard Ball

Consider a nice hyperbolic dynamical system (singularities not excluded). Statements about the topological smallness of the subset of orbits, which avoid an open subset of the phase space (for every moment of time, or just for a not too small subset of times), play a key role in showing hyperbolicity or ergodicity of semi-dispersive billiards, especially, of hard-ball systems. As well as surveying the characteristic results, called ball-avoiding theorems, and giving an idea of the methods of their proofs, their applications are also illustrated. Furthermore, we also discuss analogous questions (which had arisen, for instance, in number theory), when the Hausdorff dimension is taken instead of the topological one. The answers strongly depend on the notion of dimension which is used. Finally, ball-avoiding subsets are naturally related to repellers extensively studied by physicists. For the interested reader we also sketch some analytical and rigorous results about repellers and escape times.

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Multifractal analysis of Birkhoff averages for countable Markov maps

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2015.44 ◽

2015 ◽

Vol 35 (8) ◽

pp. 2559-2586 ◽

Cited By ~ 11

Author(s):

GODOFREDO IOMMI ◽

THOMAS JORDAN

Keyword(s):

Number Theory ◽

Hausdorff Dimension ◽

Multifractal Analysis ◽

Multifractal Formalism ◽

Interval Maps ◽

Real Analytic ◽

New Phenomena ◽

Set Of Points

In this paper we prove a multifractal formalism of Birkhoff averages for interval maps with countably many branches. Furthermore, we prove that under certain assumptions the Birkhoff spectrum is real analytic. We also show that new phenomena occur; indeed, the spectrum can be constant or it can have points where it is not analytic. Conditions for these to happen are obtained. Applications of these results to number theory are also given. Finally, we compute the Hausdorff dimension of the set of points for which the Birkhoff average is infinite.

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Non-dense orbits on homogeneous spaces and applications to geometry and number theory

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2021.4 ◽

2021 ◽

pp. 1-46

Author(s):

JINPENG AN ◽

LIFAN GUAN ◽

DMITRY KLEINBOCK

Keyword(s):

Number Theory ◽

Hausdorff Dimension ◽

Symmetric Spaces ◽

Homogeneous Spaces ◽

Discrete Subgroup ◽

Locally Symmetric Spaces ◽

Affine Map ◽

Integer Points ◽

Dense Orbits ◽

Set Of Points

Abstract Let G be a Lie group, let $\Gamma \subset G$ be a discrete subgroup, let $X=G/\Gamma $ and let f be an affine map from X to itself. We give conditions on a submanifold Z of X that guarantee that the set of points $x\in X$ with f-trajectories avoiding Z is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on X. This has applications in constructing exceptional geodesics on locally symmetric spaces and in non-density of the set of values of certain functions at integer points.

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