Stull, Donald M.
Results on the Dimension Spectra of Planar Lines
Abstract
In this paper we investigate the (effective) dimension spectra of lines in the Euclidean plane. The dimension spectrum of a line L_{a,b}, sp(L), with slope a and intercept b is the set of all effective dimensions of the points (x, ax + b) on L. It has been recently shown that, for every a and b with effective dimension less than 1, the dimension spectrum of L_{a,b} contains an interval. Our first main theorem shows that this holds for every line. Moreover, when the effective dimension of a and b is at least 1, sp(L) contains a unit interval.
Our second main theorem gives lower bounds on the dimension spectra of lines. In particular, we show that for every alpha in [0,1], with the exception of a set of Hausdorff dimension at most alpha, the effective dimension of (x, ax + b) is at least alpha + dim(a,b)/2. As a consequence of this theorem, using a recent characterization of Hausdorff dimension using effective dimension, we give a new proof of a result by Molter and Rela on the Hausdorff dimension of Furstenberg sets.
BibTeX  Entry
@InProceedings{stull:LIPIcs:2018:9661,
author = {Donald M. Stull},
title = {{Results on the Dimension Spectra of Planar Lines}},
booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
pages = {79:179:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770866},
ISSN = {18688969},
year = {2018},
volume = {117},
editor = {Igor Potapov and Paul Spirakis and James Worrell},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9661},
URN = {urn:nbn:de:0030drops96611},
doi = {10.4230/LIPIcs.MFCS.2018.79},
annote = {Keywords: algorithmic randomness, geometric measure theory, Hausdorff dimension, Kolmogorov complexity}
}
27.08.2018
Keywords: 

algorithmic randomness, geometric measure theory, Hausdorff dimension, Kolmogorov complexity 
Seminar: 

43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Issue date: 

2018 
Date of publication: 

27.08.2018 