# Lund University Publications

## LUND UNIVERSITY LIBRARIES

(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}.
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)
2016-04-01 14:00:13
date last changed
2021-04-13 05:10:08
```@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},