Bounded Lüroth expansions: applying Schmidt games where infinite distortion exists

Bill Mance; Jimmy Tseng

Acta Arithmetica (2013)

  • Volume: 158, Issue: 1, page 33-47
  • ISSN: 0065-1036

Abstract

top
We show that the set of numbers with bounded Lüroth expansions (or bounded Lüroth series) is winning and strong winning. From either winning property, it immediately follows that the set is dense, has full Hausdorff dimension, and satisfies a countable intersection property. Our result matches the well-known analogous result for bounded continued fraction expansions or, equivalently, badly approximable numbers. We note that Lüroth expansions have a countably infinite Markov partition, which leads to the notion of infinite distortion (in the sense of Markov partitions).

How to cite

top

Bill Mance, and Jimmy Tseng. "Bounded Lüroth expansions: applying Schmidt games where infinite distortion exists." Acta Arithmetica 158.1 (2013): 33-47. <http://eudml.org/doc/278948>.

@article{BillMance2013,
abstract = { We show that the set of numbers with bounded Lüroth expansions (or bounded Lüroth series) is winning and strong winning. From either winning property, it immediately follows that the set is dense, has full Hausdorff dimension, and satisfies a countable intersection property. Our result matches the well-known analogous result for bounded continued fraction expansions or, equivalently, badly approximable numbers. We note that Lüroth expansions have a countably infinite Markov partition, which leads to the notion of infinite distortion (in the sense of Markov partitions). },
author = {Bill Mance, Jimmy Tseng},
journal = {Acta Arithmetica},
keywords = {bounded Lüroth expansions; badly approximable numbers, Schmidt games; Hausdorff dimension; infinite Markov partitions},
language = {eng},
number = {1},
pages = {33-47},
title = {Bounded Lüroth expansions: applying Schmidt games where infinite distortion exists},
url = {http://eudml.org/doc/278948},
volume = {158},
year = {2013},
}

TY - JOUR
AU - Bill Mance
AU - Jimmy Tseng
TI - Bounded Lüroth expansions: applying Schmidt games where infinite distortion exists
JO - Acta Arithmetica
PY - 2013
VL - 158
IS - 1
SP - 33
EP - 47
AB - We show that the set of numbers with bounded Lüroth expansions (or bounded Lüroth series) is winning and strong winning. From either winning property, it immediately follows that the set is dense, has full Hausdorff dimension, and satisfies a countable intersection property. Our result matches the well-known analogous result for bounded continued fraction expansions or, equivalently, badly approximable numbers. We note that Lüroth expansions have a countably infinite Markov partition, which leads to the notion of infinite distortion (in the sense of Markov partitions).
LA - eng
KW - bounded Lüroth expansions; badly approximable numbers, Schmidt games; Hausdorff dimension; infinite Markov partitions
UR - http://eudml.org/doc/278948
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.