On a problem of K. Mahler: Diophantine approximation and Cantor sets

Levesley, Jeremy; Salp, Cem; Velani, Sanju

doi:10.1007/s00208-006-0069-8

Cited by 71 publications

(141 citation statements)

References 17 publications

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“…In addition, we demonstrate a correspondence between the above case and the case considered by Levesley, Salp, and Velani [9], who approximated points in the Cantor set by the left endpoints of the Cantor set. By translating their results into our setting we can prove the following theorem: Theorem 3.10.…”

Section: Corollary For Every Irrational X ∈ Rsupporting

confidence: 56%

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 ∈ Rsupporting

confidence: 56%

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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“…Similar results were obtained in R d for various families {(x n , l n )} n≥1 [8,9,1,23] or in metric spaces enjoying enough self-similar properties to support a monofractal measure [13,14,4], like the middle third Cantor set [26].…”

Section: Introductionsupporting

confidence: 78%

Ubiquity and large intersections properties under digit frequencies constraints

Barral¹,

Seuret²

2008

Math. Proc. Camb. Phil. Soc.

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We are interested in two properties of real numbers: the first one is the property of being well-approximated by some dense family of real numbers {x n } n≥1 , such as rational numbers and more generally algebraic numbers, and the second one is the property of having given digit frequencies in some b-adic expansion.We combine these two ways of classifying the real numbers, in order to provide a finer classification. We exhibit sets S of points x which are approximated at a given rate by some of the {x n } n , those x n being selected according to their digit frequencies. We compute the Hausdorff dimension of any countable intersection of such sets S, and prove that these sets enjoy the so-called large intersection property.

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“…However, if v is sufficiently large -precisely, if v ≥ ( √ 5 + 1)/2 -, a simple argument based on triangle inequalities (see e.g. Section 8 of [21]) implies (3.1).…”

Section: Approximation To Points In the Middle Third Cantor Setmentioning

confidence: 99%

“…Apparently, the methods used in [21] do not allow us to replace the ≥ sign by the = sign in the left hand side of (3.2). They, however, give the following slight refinement of (3.2):…”

Section: Approximation To Points In the Middle Third Cantor Setmentioning

confidence: 99%