(Preliminary Version)
  • David Doty
  • Xiaoyang Gu
  • Jack H. Lutz
  • Elvira Mayordomo
  • Philippe Moser
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3618)


The zeta-dimension of a set A of positive integers is

Dim ζ (A) = inf{s | ζ A (s) < ∞ },


\(\zeta_A(s)=\sum_{n\in A}n^{-s}.\)

Zeta-dimension serves as a fractal dimension on ℤ +  that extends naturally and usefully to discrete lattices such as ℤ d , where d is a positive integer.

This paper reviews the origins of zeta-dimension (which date to the eighteenth and nineteenth centuries) and develops its basic theory, with particular attention to its relationship with algorithmic information theory. New results presented include a gale characterization of zeta-dimension and a theorem on the zeta-dimensions of pointwise sums and products of sets of positive integers.


Positive Integer Fractal Dimension Cellular Automaton Dirichlet Series Kolmogorov Complexity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • David Doty
    • 1
  • Xiaoyang Gu
    • 1
  • Jack H. Lutz
    • 1
  • Elvira Mayordomo
    • 2
  • Philippe Moser
    • 1
  1. 1.Department of Computer ScienceIowa State UniversityAmesUSA
  2. 2.Departamento de Informática e Ingeniería de SistemasUniversidad de ZaragozaZaragozaSpain

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