Metric Diophantine approximation on the middle-third Cantor set

Yann Bugeaud; Arnaud Durand

Journal of the European Mathematical Society (2016)

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

Abstract

top
Let μ 2 be a real number and let ( μ ) denote the set of real numbers approximable at order at least μ by rational numbers. More than eighty years ago, Jarník and, independently, Besicovitch established that the Hausdorff dimension of ( μ ) is equal to 2 / μ . We investigate the size of the intersection of ( μ ) with Ahlfors regular compact subsets of the interval [ 0 , 1 ] . In particular, we propose a conjecture for the exact value of the dimension of ( μ ) 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.

How to cite

top

Bugeaud, 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 ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.