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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}.
Please use this url to cite or link to this publication:
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
2016-04-01 14:00:13
date last changed
2022-01-27 22:17:06
@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}},
  doi          = {{10.1007/s11856-009-0042-9}},
  volume       = {{171}},
  year         = {{2009}},
}