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On numbers badly approximable by dyadic rationals

Nilsson, Johan LU (2009) In Israel Journal of Mathematics 171(1). p.93-110
Abstract
We consider a problem originating both from circle coverings and badly approximable numbers in the case of dyadic diophantine approximation. For the unit circle S we give an elementary proof that the set {x is an element of S : 2(n)x >= c (mod 1) n >= 0} is a fractal set whose Hausdorff dimension depends continuously on c and is constant on intervals which form a set of Lebesgue measure 1. Hence it has a fractal graph. We completely characterize the intervals where the dimension remains unchanged. As a consequence we can describe the graph of c bar right arrow dim(H) {x is an element of [0, 1] : x - m/2(n) < c/2(n) (mod 1) finitely often}.
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Israel Journal of Mathematics
volume
171
issue
1
pages
93 - 110
publisher
Hebrew University Magnes Press
external identifiers
  • wos:000267887400007
  • scopus:77749319345
ISSN
0021-2172
DOI
10.1007/s11856-009-0042-9
language
English
LU publication?
yes
id
af2dd1ce-4a62-4fdf-8820-fcdbd1438585 (old id 1462532)
date added to LUP
2009-08-19 12:36:48
date last changed
2017-08-20 04:02:35
@article{af2dd1ce-4a62-4fdf-8820-fcdbd1438585,
  abstract     = {We consider a problem originating both from circle coverings and badly approximable numbers in the case of dyadic diophantine approximation. For the unit circle S we give an elementary proof that the set {x is an element of S : 2(n)x &gt;= c (mod 1) n &gt;= 0} is a fractal set whose Hausdorff dimension depends continuously on c and is constant on intervals which form a set of Lebesgue measure 1. Hence it has a fractal graph. We completely characterize the intervals where the dimension remains unchanged. As a consequence we can describe the graph of c bar right arrow dim(H) {x is an element of [0, 1] : x - m/2(n) &lt; c/2(n) (mod 1) finitely often}.},
  author       = {Nilsson, Johan},
  issn         = {0021-2172},
  language     = {eng},
  number       = {1},
  pages        = {93--110},
  publisher    = {Hebrew University Magnes Press},
  series       = {Israel Journal of Mathematics},
  title        = {On numbers badly approximable by dyadic rationals},
  url          = {http://dx.doi.org/10.1007/s11856-009-0042-9},
  volume       = {171},
  year         = {2009},
}