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.2017.58
URN: urn:nbn:de:0030-drops-80875
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/8087/
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Lutz, Neil

Fractal Intersections and Products via Algorithmic Dimension

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LIPIcs-MFCS-2017-58.pdf (0.5 MB)

Abstract

Algorithmic dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions of intersections and Cartesian products of fractals in Euclidean spaces. This approach shows that a known intersection formula for Borel sets holds for arbitrary sets, and it significantly simplifies the proof of a known product formula. Both of these formulas are prominent, fundamental results in fractal geometry that are taught in typical undergraduate courses on the subject.

BibTeX - Entry

@InProceedings{lutz:LIPIcs:2017:8087,
  author =	{Neil Lutz},
  title =	{{Fractal Intersections and Products via Algorithmic Dimension}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{58:1--58:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Kim G. Larsen and Hans L. Bodlaender and Jean-Francois Raskin},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/8087},
  URN =		{urn:nbn:de:0030-drops-80875},
  doi =		{10.4230/LIPIcs.MFCS.2017.58},
  annote =	{Keywords: algorithmic randomness, geometric measure theory, Hausdorff dimension, Kolmogorov complexity}
}

Keywords: algorithmic randomness, geometric measure theory, Hausdorff dimension, Kolmogorov complexity
Collection: 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)
Issue Date: 2017
Date of publication: 01.12.2017

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