# Metric Diophantine approximation on the middle-third Cantor set

Journal of the European Mathematical Society (2016)

- Volume: 018, Issue: 6, page 1233-1272
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topBugeaud, Yann, and Durand, Arnaud. "Metric Diophantine approximation on the middle-third Cantor set." Journal of the European Mathematical Society 018.6 (2016): 1233-1272. <http://eudml.org/doc/277340>.

@article{Bugeaud2016,

abstract = {Let $\mu \ge 2$ be a real number and let $\mathcal \{M\} (\mu )$ denote the set of real numbers approximable at order at least $\mu $ by rational numbers. More than eighty years ago, Jarník and, independently, Besicovitch established that the Hausdorff dimension of $\mathcal \{M\} (\mu )$ is equal to $2 / \mu $. We investigate the size of the intersection of $\mathcal \{M\} (\mu )$ with Ahlfors regular compact subsets of the interval $[0, 1]$. In particular, we propose a conjecture for the exact value of the dimension of $\mathcal \{M\} (\mu )$ intersected with the middle-third Cantor set and give several results supporting this conjecture. We show in particular that the conjecture holds for a natural probabilistic model that is intended to mimic the distribution of the rationals. The core of our study relies heavily on dimension estimates concerning the set of points lying in an Ahlfors regular set and approximated at a given rate by a system of random points.},

author = {Bugeaud, Yann, Durand, Arnaud},

journal = {Journal of the European Mathematical Society},

keywords = {Diophantine approximation; Hausdorff dimension; irrationality exponent; Cantor set; Mahler’s problem; Diophantine approximation; Hausdorff dimension; irrationality exponent; Cantor set; Mahler's problem},

language = {eng},

number = {6},

pages = {1233-1272},

publisher = {European Mathematical Society Publishing House},

title = {Metric Diophantine approximation on the middle-third Cantor set},

url = {http://eudml.org/doc/277340},

volume = {018},

year = {2016},

}

TY - JOUR

AU - Bugeaud, Yann

AU - Durand, Arnaud

TI - Metric Diophantine approximation on the middle-third Cantor set

JO - Journal of the European Mathematical Society

PY - 2016

PB - European Mathematical Society Publishing House

VL - 018

IS - 6

SP - 1233

EP - 1272

AB - Let $\mu \ge 2$ be a real number and let $\mathcal {M} (\mu )$ denote the set of real numbers approximable at order at least $\mu $ by rational numbers. More than eighty years ago, Jarník and, independently, Besicovitch established that the Hausdorff dimension of $\mathcal {M} (\mu )$ is equal to $2 / \mu $. We investigate the size of the intersection of $\mathcal {M} (\mu )$ with Ahlfors regular compact subsets of the interval $[0, 1]$. In particular, we propose a conjecture for the exact value of the dimension of $\mathcal {M} (\mu )$ intersected with the middle-third Cantor set and give several results supporting this conjecture. We show in particular that the conjecture holds for a natural probabilistic model that is intended to mimic the distribution of the rationals. The core of our study relies heavily on dimension estimates concerning the set of points lying in an Ahlfors regular set and approximated at a given rate by a system of random points.

LA - eng

KW - Diophantine approximation; Hausdorff dimension; irrationality exponent; Cantor set; Mahler’s problem; Diophantine approximation; Hausdorff dimension; irrationality exponent; Cantor set; Mahler's problem

UR - http://eudml.org/doc/277340

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.