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Results on the Dimension Spectra of Planar Lines

Author Donald M. Stull

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ACM Subject Classification
  • Theory of computation → Complexity theory and logic
Keywords
  • algorithmic randomness
  • geometric measure theory
  • Hausdorff dimension
  • Kolmogorov complexity

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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.

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Donald M. Stull. Results on the Dimension Spectra of Planar Lines. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 79:1-79:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.MFCS.2018.79

Author Details

Donald M. Stull
  • Inria Nancy-Grand Est, 615 rue du jardin botanique, 54600 Villers-les-Nancy, France

References

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