License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2018.79
URN: urn:nbn:de:0030-drops-96611
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9661/
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Stull, Donald M.

Results on the Dimension Spectra of Planar Lines

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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 =	{79:1--79:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Igor Potapov and Paul Spirakis and James Worrell},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9661},
  URN =		{urn:nbn:de:0030-drops-96611},
  doi =		{10.4230/LIPIcs.MFCS.2018.79},
  annote =	{Keywords: algorithmic randomness, geometric measure theory, Hausdorff dimension, Kolmogorov complexity}
}

Keywords: algorithmic randomness, geometric measure theory, Hausdorff dimension, Kolmogorov complexity
Collection: 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)
Issue Date: 2018
Date of publication: 27.08.2018

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