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Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions

Li, Bing; Persson, Tomas LU ; Wang, Baowei and Wu, Jun (2014) In Mathematische Zeitschrift 276(3-4). p.799-827
Abstract
We consider the distribution of the orbits of the number 1 under the -transformations as varies. Mainly, the size of the set of for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension. The dimension of the following set is determined, where is a given point in and is a sequence of integers tending to infinity as . For the proof of this result, the notion of the recurrence time of a word in symbolic space is introduced to characterise the lengths and the distribution of cylinders (the set of with a common prefix in the expansion of 1) in the parameter space .
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
beta-expansion, Diophantine approximation, Hausdorff dimension
in
Mathematische Zeitschrift
volume
276
issue
3-4
pages
799 - 827
publisher
Springer
external identifiers
  • wos:000332835800009
  • scopus:84897647951
ISSN
0025-5874
DOI
10.1007/s00209-013-1223-0
language
English
LU publication?
yes
id
fa8f996e-2aab-4ba2-ba55-e3563c0365b5 (old id 4410957)
date added to LUP
2014-04-29 11:20:58
date last changed
2017-09-24 03:03:44
@article{fa8f996e-2aab-4ba2-ba55-e3563c0365b5,
  abstract     = {We consider the distribution of the orbits of the number 1 under the -transformations as varies. Mainly, the size of the set of for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension. The dimension of the following set is determined, where is a given point in and is a sequence of integers tending to infinity as . For the proof of this result, the notion of the recurrence time of a word in symbolic space is introduced to characterise the lengths and the distribution of cylinders (the set of with a common prefix in the expansion of 1) in the parameter space .},
  author       = {Li, Bing and Persson, Tomas and Wang, Baowei and Wu, Jun},
  issn         = {0025-5874},
  keyword      = {beta-expansion,Diophantine approximation,Hausdorff dimension},
  language     = {eng},
  number       = {3-4},
  pages        = {799--827},
  publisher    = {Springer},
  series       = {Mathematische Zeitschrift},
  title        = {Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions},
  url          = {http://dx.doi.org/10.1007/s00209-013-1223-0},
  volume       = {276},
  year         = {2014},
}