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Fractal Intersections and Products via Algorithmic Dimension

Author Neil Lutz



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Neil Lutz

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Neil Lutz. Fractal Intersections and Products via Algorithmic Dimension. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 58:1-58:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.MFCS.2017.58

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.
Keywords
  • algorithmic randomness
  • geometric measure theory
  • Hausdorff dimension
  • Kolmogorov complexity

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