On numbers badly approximable by dyadic rationals
(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:
https://lup.lub.lu.se/record/1462532
- author
- Nilsson, Johan LU
- organization
- publishing date
- 2009
- 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 >= 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}.}}, 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}}, }