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.2018.71
URN: urn:nbn:de:0030-drops-96532
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9653/
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Lutz, Neil ; Stull, Donald M.

Projection Theorems Using Effective Dimension

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Abstract

In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrand's projection theorem, which shows that for every analytic set E, for almost every line L, the Hausdorff dimension of the orthogonal projection of E onto L is maximal.
We use Kolmogorov complexity to give two new results on the Hausdorff and packing dimensions of orthogonal projections onto lines. The first shows that the conclusion of Marstrand's theorem holds whenever the Hausdorff and packing dimensions agree on the set E, even if E is not analytic. Our second result gives a lower bound on the packing dimension of projections of arbitrary sets. Finally, we give a new proof of Marstrand's theorem using the theory of computing.

BibTeX - Entry

@InProceedings{lutz_et_al:LIPIcs:2018:9653,
  author =	{Neil Lutz and Donald M. Stull},
  title =	{{Projection Theorems Using Effective Dimension}},
  booktitle =	{43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018)},
  pages =	{71:1--71:15},
  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/2018/9653},
  URN =		{urn:nbn:de:0030-drops-96532},
  doi =		{10.4230/LIPIcs.MFCS.2018.71},
  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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