On numbers badly approximable by dyadic rationals
(2009) In Israel Journal of Mathematics 171(1). p.93110 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:
http://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
 00212172
 DOI
 10.1007/s1185600900429
 language
 English
 LU publication?
 yes
 id
 af2dd1ce4a624fdf8820fcdbd1438585 (old id 1462532)
 date added to LUP
 20090819 12:36:48
 date last changed
 20170820 04:02:35
@article{af2dd1ce4a624fdf8820fcdbd1438585, 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 = {00212172}, language = {eng}, number = {1}, pages = {93110}, 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/s1185600900429}, volume = {171}, year = {2009}, }