Schmidt’s game on fractals

Fishman, Lior

doi:10.1007/s11856-009-0041-x

Cited by 41 publications

(63 citation statements)

References 6 publications

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“…This fact demonstrates that this approximation function cannot be improved for almost all irrationals. We remark that the subject of approximating points on fractals by rationals has been extensively studied in recent years; see for example [3,7,8] for badly approximable numbers and [6,14] for very well approximable numbers. In [2], elements of the Cantor set with any prescribed irrationality exponent were explicitly constructed.…”

Section: Corollary For Every Irrational X ∈ Rmentioning

confidence: 99%

Intrinsic approximation for fractals defined by rational iterated function systems: Mahler's research suggestion

Fishman

Simmons

2014

Proceedings of the London Mathematical Society

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Abstract. In this paper, we consider intrinsic Diophantine approximation in the sense of K. Mahler (1984) on the Cantor set and similar fractals. We begin by obtaining a Dirichlet type theorem for the limit set of a rational iterated function system. Next, we investigate the rigidity of this result by applying a random affine transformation to such a fractal and determining the intrinsic Diophantine theory of the image fractal. The final two sections concern the optimality of the Dirichlet type theorem established at the beginning. The first of these seeks to show optimality in the sense that any proof using the same method as ours cannot prove a better approximation exponent, in a precise sense. This is done by introducing a new height function on the rationals intrinsic to the fractal and studying the Diophantine properties of points on the fractal with respect to this new height function. In the final section, we use a result of S. Ramanujan to give a lower bound on the periods of rationals which could cause exceptions to the optimality of the approximation exponent (this time with the usual height function). We give a heuristic argument suggesting that there are only finitely many rationals with periods so large; if this is true, then the approximation exponent is optimal for the Cantor set.

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Section: Corollary For Every Irrational X ∈ Rmentioning

confidence: 99%

Intrinsic approximation for fractals defined by rational iterated function systems: Mahler's research suggestion

Fishman

Simmons

2014

Proceedings of the London Mathematical Society

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“…where C is an absolute constant. It was proven in [7] that if J ⊆ R is any Ahlfors regular set, then the union BA 1 def = ε>0 BA 1 (ε) has full dimension in J. We can prove the following quantitative version of this result: 5 We estimated the dimension of F 2 ∩C by searching for a disjoint cover of F 2 by intervals of the form Iω = [[0; ω, 1], [0; ω, 3]] or Iω = [[0; ω, 3], [0; ω, 1]] (see (6.7) for the notation), where ω is a finite word in the alphabet {1, 2}, such that either…”

Section: Resultsmentioning

confidence: 95%

“…(This follows from the corresponding property for classical winning sets, see e.g. [7,Theorem 3.1].) The following lemma is a quantitative version of this property: Lemma 3.7.…”

Section: The Absolute Game and Its Applicationsmentioning

confidence: 96%

Quantitative results using variants of Schmidt’s game: Dimension bounds, arithmetic progressions, and more

2019

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Schmidt's game is generally used to deduce qualitative information about the Hausdorff dimensions of fractal sets and their intersections. However, one can also ask about quantitative versions of the properties of winning sets. In this paper we show that such quantitative information has applications to various questions including:• What is the maximal length of an arithmetic progression on the "middle ε" Cantor set?• What is the smallest n such that there is some element of the ternary Cantor set whose continued fraction partial quotients are all ≤ n? • What is the Hausdorff dimension of the set of ε-badly approximable numbers on the Cantor set? We show that a variant of Schmidt's game known as the potential game is capable of providing better bounds on the answers to these questions than the classical Schmidt's game. We also use the potential game to provide a new proof of an important lemma in the classical proof of the existence of Hall's Ray.

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“…Namely, a result of Mayeda and Merrill states that the set

{Bad}_{y}

is winning for a different game introduced by McMullen and known as the ‘absolute game’, while the results of Broderick, Fishman, Kleinbock, Reich, and Weiss state that any set winning for the absolute game can be intersected with any sufficiently nice fractal (and in particular any Ahlfors

δ

‐regular fractal) to get a set winning for Schmidt's game (played on that fractal). This immediately implies that

K \cap {boldBad}_{y}

is winning for Schmidt's game (played on

K

) and according to a theorem of Fishman this implies the existence of the family of sets

{F_{s} : s < δ}

described above, and in particular that

dim (K \cap {boldBad}_{y}) ⩾ δ

. To make this paper more self‐contained, in what follows we give the details behind this argument, as well as recalling the definition of Schmidt's game and the absolute game.…”

Section: Proof Of Theoremmentioning

confidence: 94%

Diophantine approximation in Kleinian groups: singular, extremal, and bad limit points

Beresnevich

Ghosh

Simmons

et al. 2018

J. London Math. Soc.

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The overall aim of this note is to initiate a "manifold" theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite Kleinian group framework. Also, we consider the natural analogue of Davenport's problem regarding badly approximable limit points in a given subset of the limit set. Beyond extremality, we discuss potential Khintchine-type statements for subsets of the limit set. These can be interpreted as the conjectural "manifold" strengthening of Sullivan's logarithmic law for geodesics. The general setup and main problemsThe classical results of Diophantine approximation, in particular those from the onedimensional theory, have natural counterparts and extensions in the hyperbolic space setting. In this setting, instead of approximating real numbers by rationals, one approximates the limit points of a fixed Kleinian group G by points in the orbit (under the group) of a distinguished limit point y. Beardon & Maskit [4] have shown that the geometry of the group is reflected in the approximation properties of points in the limit set.Unless stated otherwise, in what follows G denotes a nonelementary, geometrically finite Kleinian group acting on the unit ball model (B d+1 , ρ) of (d + 1)-dimensional hyperbolic space with metric ρ derived from the differential dρ = 2|dx|/(1−|x| 2 ). Thus, G is a discrete subgroup of Möb(B d+1 ), the group of orientation-preserving Möbius *

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Schmidt’s game on fractals

Cited by 41 publications

References 6 publications

Intrinsic approximation for fractals defined by rational iterated function systems: Mahler's research suggestion

Intrinsic approximation for fractals defined by rational iterated function systems: Mahler's research suggestion

Quantitative results using variants of Schmidt’s game: Dimension bounds, arithmetic progressions, and more

Diophantine approximation in Kleinian groups: singular, extremal, and bad limit points

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