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The Dimension Spectrum Conjecture for Planar Lines

Author D. M. Stull

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Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • Algorithmic randomness
  • Kolmogorov complexity
  • effective dimension

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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].

Cite As Get BibTex

D. M. Stull. The Dimension Spectrum Conjecture for Planar Lines. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 133:1-133:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.133

Author Details

D. M. Stull
  • Department of Computer Science, Northwestern University, Evanston, IL, USA

References

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