The Dimension of a Point: Computability Meets Fractal Geometry

  • Jack H. Lutz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3526)


Recent developments in the theory of computing give a canonical way of assigning a dimension to each point of n-dimensional Euclidean space. Computable points have dimension 0, random points have dimension n, and every real number in [0,n] is the dimension of uncountably many points. If X is a reasonably simple subset of n-dimensional Euclidean space (a union of computably closed sets), then the classical Hausdorff dimension of X is just the supremum of the dimensions of the points in X. In this talk I will discuss the meaning of these developments, their implications for both the theory of computing and fractal geometry, and directions for future research.


  1. 1.
    Falconer, K.: The Geometry of Fractal Sets. Cambridge University Press, Cambridge (1985)zbMATHCrossRefGoogle Scholar
  2. 2.
    Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Chichester (1990)zbMATHGoogle Scholar
  3. 3.
    Hitchcock, J.M.: Correspondence principles for effective dimensions. Theory of Computing Systems. In: Proceedings of the 29th International Colloquium on Automata, Languages, and Programming, pp. 561–571 (2002) (to appear) (Preliminary version appeared) Google Scholar
  4. 4.
    Lutz, J.H.: Dimension in complexity classes. SIAM Journal on Computing 32, 1236–1259 (2003); Preliminary version appeared in Proceedings of the Fifteenth Annual IEEE Conference on Computational Complexity, pp. 158–169 (2000)Google Scholar
  5. 5.
    Lutz, J.H.: The dimensions of individual strings and sequences. Information and Computation 187, 49–79 (2003)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Jack H. Lutz
    • 1
  1. 1.Department of Computer ScienceIowa State UniversityAmesUSA

Personalised recommendations