Dimension Spectra of Lines

  • Neil Lutz
  • D. M. Stull
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10307)


This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum \({{\mathrm{sp}}}(L)\) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension \(\dim (a, b)\) is equal to the effective packing dimension \({{\mathrm{Dim}}}(a, b)\), then \({{\mathrm{sp}}}(L)\) contains a unit interval. We also show that, if the dimension \(\dim (a, b)\) is at least one, then \({{\mathrm{sp}}}(L)\) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.


  1. 1.
    Athreya, K.B., Hitchcock, J.M., Lutz, J.H., Mayordomo, E.: Effective strong dimension in algorithmic information and computational complexity. SIAM J. Comput. 37(3), 671–705 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Dougherty, R., Lutz, J., Mauldin, R.D., Teutsch, J.: Translating the Cantor set by a random real. Trans. Am. Math. Soc. 366(6), 3027–3041 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity. Springer, New York (2010)CrossRefMATHGoogle Scholar
  4. 4.
    Xiaoyang, G., Lutz, J.H., Mayordomo, E., Moser, P.: Dimension spectra of random subfractals of self-similar fractals. Ann. Pure Appl. Logic 165(11), 1707–1726 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Hitchcock, J.M.: Correspondence principles for effective dimensions. Theory Comput. Syst. 38(5), 559–571 (2005)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Li, M., Vitányi, P.M.B.: An Introduction to Kolmogorov Complexity and Its Applications, 3rd edn. Springer, New York (2008)CrossRefMATHGoogle Scholar
  7. 7.
    Lutz, J.H.: The dimensions of individual strings and sequences. Inf. Comput. 187(1), 49–79 (2003)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Lutz, J.H., Lutz, N.: Algorithmic information, plane Kakeya sets, and conditional dimension. In: Vollmer, H., Vallee, B. (eds.) 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017), Leibniz International Proceedings in Informatics (LIPIcs), vol. 66, pp. 53:1–53:13. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Germany (2017)Google Scholar
  9. 9.
    Lutz, J.H., Mayordomo, E.: Dimensions of points in self-similar fractals. SIAM J. Comput. 38(3), 1080–1112 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lutz, N., Stull, D.M.: Bounding the dimension of points on a line. In: Gopal, T.V., Jäger, G., Steila, S. (eds.) TAMC 2017. LNCS, vol. 10185, pp. 425–439. Springer, Cham (2017). doi: 10.1007/978-3-319-55911-7_31 CrossRefGoogle Scholar
  11. 11.
    Mayordomo, E.: A Kolmogorov complexity characterization of constructive Hausdorff dimension. Inf. Process. Lett. 84(1), 1–3 (2002)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Reimann, J.: Effectively closed classes of measures and randomness. Ann. Pure Appl. Logic 156(1), 170–182 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Turetsky, D.: Connectedness properties of dimension level sets. Theor. Comput. Sci. 412(29), 3598–3603 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Westrick, L.B.: Computability in ordinal ranks and symbolic dynamics. Ph.D. thesis, University of California, Berkeley (2014)Google Scholar
  15. 15.
    Wolff, T.: Recent work connected with the Kakeya problem. In: Prospects in Mathematics, pp. 129–162 (1999)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceRutgers UniversityPiscatawayUSA
  2. 2.Department of Computer ScienceIowa State UniversityAmesUSA

Personalised recommendations