License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2022.133
URN: urn:nbn:de:0030-drops-164749
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16474/
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Stull, D. M.

The Dimension Spectrum Conjecture for Planar Lines

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LIPIcs-ICALP-2022-133.pdf (0.7 MB)

Abstract

Let L_{a,b} be a line in the Euclidean plane with slope a and intercept b. The dimension spectrum sp(L_{a,b}) is the set of all effective dimensions of individual points on L_{a,b}. Jack Lutz, in the early 2000s posed the dimension spectrum conjecture. This conjecture states that, for every line L_{a,b}, the spectrum of L_{a,b} contains a unit interval.
In this paper we prove that the dimension spectrum conjecture is true. Specifically, let (a,b) be a slope-intercept pair, and let d = min{dim(a,b), 1}. For every s ∈ [0, 1], we construct a point x such that dim(x, ax + b) = d + s. Thus, we show that sp(L_{a,b}) contains the interval [d, 1+ d].

BibTeX - Entry

@InProceedings{stull:LIPIcs.ICALP.2022.133,
  author =	{Stull, D. M.},
  title =	{{The Dimension Spectrum Conjecture for Planar Lines}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{133:1--133:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16474},
  URN =		{urn:nbn:de:0030-drops-164749},
  doi =		{10.4230/LIPIcs.ICALP.2022.133},
  annote =	{Keywords: Algorithmic randomness, Kolmogorov complexity, effective dimension}
}

Keywords: Algorithmic randomness, Kolmogorov complexity, effective dimension
Collection: 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)
Issue Date: 2022
Date of publication: 28.06.2022

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