Document Open Access Logo

Results on the Dimension Spectra of Planar Lines

Author Donald M. Stull

Thumbnail PDF


  • Filesize: 412 kB
  • 15 pages

Document Identifiers

Author Details

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

Cite AsGet BibTex

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)


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.

Subject Classification

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


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Krishna B. Athreya, John M. Hitchcock, Jack H. Lutz, and Elvira Mayordomo. Effective strong dimension in algorithmic information and computational complexity. SIAM J. Comput., 37(3):671-705, 2007. Google Scholar
  2. Adam Case and Jack H. Lutz. Mutual dimension. ACM Transactions on Computation Theory, 7(3):12, 2015. Google Scholar
  3. Jack H. Lutz. The dimensions of individual strings and sequences. Inf. Comput., 187(1):49-79, 2003. Google Scholar
  4. Jack H. Lutz and Neil Lutz. Algorithmic information, plane Kakeya sets, and conditional dimension. ACM Transactions on Computation Theory, to appear. Google Scholar
  5. Jack H. Lutz and Elvira Mayordomo. Dimensions of points in self-similar fractals. SIAM J. Comput., 38(3):1080-1112, 2008. Google Scholar
  6. Neil Lutz. Fractal intersections and products via algorithmic dimension. In 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017, August 21-25, 2017 - Aalborg, Denmark, pages 58:1-58:12, 2017. Google Scholar
  7. Neil Lutz. Some open problems in algorithmic fractal geometry. Open Problem Column, SIGACT News, 48(4), 2017. Google Scholar
  8. Neil Lutz and DM Stull. Projection theorems using effective dimension. arXiv preprint arXiv:1711.02124, 2017. Google Scholar
  9. Neil Lutz and Donald M. Stull. Bounding the dimension of points on a line. In Theory and Applications of Models of Computation - 14th Annual Conference, TAMC 2017, Bern, Switzerland, April 20-22, 2017, Proceedings, pages 425-439, 2017. Google Scholar
  10. Neil Lutz and Donald M. Stull. Dimension spectra of lines. In Unveiling Dynamics and Complexity - 13th Conference on Computability in Europe, CiE 2017, Turku, Finland, June 12-16, 2017, Proceedings, pages 304-314, 2017. Google Scholar
  11. Elvira Mayordomo. A Kolmogorov complexity characterization of constructive Hausdorff dimension. Inf. Process. Lett., 84(1):1-3, 2002. Google Scholar
  12. Ursula Molter and Ezequiel Rela. Furstenberg sets for a fractal set of directions. Proc. Amer. Math. Soc., 140:2753-2765, 2012. Google Scholar
  13. Daniel Turetsky. Connectedness properties of dimension level sets. Theor. Comput. Sci., 412(29):3598-3603, 2011. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail