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Fractal Intersections and Products via Algorithmic Dimension

Published:31 August 2021Publication History
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Abstract

Algorithmic fractal 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 two prominent, fundamental results about the dimension of Borel or analytic sets also hold for arbitrary sets.

References

  1. K. B. Athreya, J. M. Hitchcock, J. H. Lutz, and E. Mayordomo. 2007. Effective strong dimension in algorithmic information and computational complexity. SIAM Journal of Computing 37, 3 (2007), 671–705. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. V. Becher, J. Reimann, and T. A. Slaman. 2018. Irrationality exponent, Hausdorff dimension and effectivization. Monatshefte für Mathematik 185, 2 (2018), 167–188.Google ScholarGoogle ScholarCross RefCross Ref
  3. Bishop, C. J. 2017. Personal communication. April 27.Google ScholarGoogle Scholar
  4. C. J. Bishop and Y. Peres. 1996. Packing dimension and Cartesian products. Transactions of the American Mathematical Society 348 (1996), 4433–4445.Google ScholarGoogle ScholarCross RefCross Ref
  5. C. J. Bishop and Y. Peres. 2016. Fractals in Probability and Analysis. Cambridge Studies in Advanced Mathematics. Cambridge University Press.Google ScholarGoogle Scholar
  6. J. Cai and J. Hartmanis. 1994. On Hausdorff and topological dimensions of the Kolmogorov complexity of the real line. Journal of Computer and System Sciences 49, 3 (1994), 605–619. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. A. Case and J. H. Lutz. 2015. Mutual dimension. ACM Transactions on Computation Theory 7, 3 (2015), 12. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. R. O. Davies. 1971. Some remarks on the Kakeya problem. Proceedings of the Cambridge Philosophical Society 69 (1971), 417–421.Google ScholarGoogle ScholarCross RefCross Ref
  9. K. J. Falconer. 1985. The Geometry of Fractal Sets. Cambridge University Press. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. K. J. Falconer. 1994. Sets with large intersection properties. Journal of the London Mathematical Society 49, 2 (1994), 267–280.Google ScholarGoogle ScholarCross RefCross Ref
  11. K. J. Falconer. 2014. Fractal Geometry: Mathematical Foundations and Applications (3rd ed.). Wiley.Google ScholarGoogle Scholar
  12. F. Hausdorff. 1919. Dimension und äusseres Mass. Mathematische Annalen 79 (1919), 157–179.Google ScholarGoogle ScholarCross RefCross Ref
  13. J.-P. Kahane. 1986. Sur la dimension des intersections. In Aspects of Mathematics and Its Applications, J. A. Barroso (Ed.). North-Holland Mathematical Library, 34. Elsevier, 419–430.Google ScholarGoogle Scholar
  14. M. Li and P. M. Vitányi. 2008. An Introduction to Kolmogorov Complexity and Its Applications (3rd ed.). Springer. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. J. H. Lutz. 2003. The dimensions of individual strings and sequences. Information and Computation 187, 1 (2003), 49–79. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. J. H. Lutz and N. Lutz. 2018. Algorithmic information, plane Kakeya sets, and conditional dimension. ACM Transactions on Computation Theory 10, 2, Article 7 (2018), 22 pages. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. J. H. Lutz and E. Mayordomo. 2008. Dimensions of points in self-similar fractals. SIAM Journal of Computing 38, 3 (2008), 1080–1112. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. N. Lutz and D. M. Stull. 2020. Bounding the dimension of points on a line. Information and Computation 275 (2020), 104601.Google ScholarGoogle ScholarCross RefCross Ref
  19. J. M. Marstrand. 1954. Some fundamental geometrical properties of plane sets of fractional dimensions. Proceedings of the London Mathematical Society 4, 3 (1954), 257–302.Google ScholarGoogle ScholarCross RefCross Ref
  20. P. Martin-Löf. 1966. The definition of random sequences. Information and Control 9, 6 (1966), 602–619.Google ScholarGoogle ScholarCross RefCross Ref
  21. P. Mattila. 1984. Hausdorff dimension and capacities of intersections of sets in -space. Acta Mathematica 152 (1984), 77–105.Google ScholarGoogle ScholarCross RefCross Ref
  22. P. Mattila. 1985. On the Hausdorff dimension and capacities of intersections. Mathematika 32 (1985), 213–217.Google ScholarGoogle ScholarCross RefCross Ref
  23. P. Mattila. 1995. Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge University Press.Google ScholarGoogle Scholar
  24. E. Mayordomo. 2002. A Kolmogorov complexity characterization of constructive Hausdorff dimension. Information Processing Letters 84, 1 (2002), 1–3. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. E. Mayordomo. 2008. Effective fractal dimension in algorithmic information theory. In New Computational Paradigms: Changing Conceptions of What Is Computable, S. B. Cooper, B. Löwe, and A. Sorbi (Eds.). Springer, New York, NY, 259–285.Google ScholarGoogle Scholar
  26. U. Molter and E. Rela. 2012. Furstenberg sets for a fractal set of directions. Proceedings of the American Mathematical Society 140 (2012), 2753–2765.Google ScholarGoogle ScholarCross RefCross Ref
  27. J. Reimann. 2004. Computability and fractal dimension. Ph.D. Thesis, Heidelberg University.Google ScholarGoogle Scholar
  28. J. Reimann. 2008. Effectively closed sets of measures and randomness. Annals of Pure and Applied Logic 156, 1 (2008), 170–182.Google ScholarGoogle ScholarCross RefCross Ref
  29. B. Ryabko. 1984. Coding of combinatorial sources and Hausdorff dimension. Soviets Mathematics Doklady 30 (1984), 219–222.Google ScholarGoogle Scholar
  30. B. Ryabko. 1986. Noiseless coding of combinatorial sources. Problems of Information Transmission 22 (1986), 170–179.Google ScholarGoogle Scholar
  31. B. Ryabko. 1993. Algorithmic approach to the prediction problem. Problems of Information Transmission 29 (1993), 186–193.Google ScholarGoogle Scholar
  32. B. Ryabko. 1994. The complexity and effectiveness of prediction algorithms. Journal of Complexity 10, 3 (1994), 281–295. Google ScholarGoogle ScholarDigital LibraryDigital Library
  33. L. Staiger. 1989. Kolmogorov complexity and Hausdorff dimension. Information and Computation 103 (1989), 159–194. Google ScholarGoogle ScholarDigital LibraryDigital Library
  34. L. Staiger. 1998. A tight upper bound on Kolmogorov complexity and uniformly optimal prediction. Theory of Computing Systems 31 (1998), 215–229.Google ScholarGoogle ScholarCross RefCross Ref
  35. E. M. Stein and R. Shakarchi. 2005. Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton Lectures in Analysis. Princeton University Press.Google ScholarGoogle ScholarCross RefCross Ref
  36. D. Sullivan. 1984. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Mathematica 153, 1 (1984), 259–277.Google ScholarGoogle ScholarCross RefCross Ref
  37. C. Tricot. 1982. Two definitions of fractional dimension. Mathematical Proceedings of the Cambridge Philosophical Society 91, 1 (1982), 57–74.Google ScholarGoogle ScholarCross RefCross Ref
  38. A. K. Zvonkin and L. A. Levin. 1970. The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms. Russian Mathematical Surveys 25 (1970), 83–124.Google ScholarGoogle ScholarCross RefCross Ref

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      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 13, Issue 3
        September 2021
        146 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/3476826
        Issue’s Table of Contents

        Copyright © 2021 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 31 August 2021
        • Revised: 1 January 2021
        • Accepted: 1 January 2021
        • Received: 1 November 2019
        Published in toct Volume 13, Issue 3

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