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Zeta-Dimension

(Preliminary Version)

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Mathematical Foundations of Computer Science 2005 (MFCS 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3618))

Abstract

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

Dim ζ (A) = inf{s | ζ A (s) < ∞ },

where

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

Zeta-dimension serves as a fractal dimension on ℤ +  that extends naturally and usefully to discrete lattices such as ℤd, 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.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Iowa State University, Ames, IA, 50011, USA

    David Doty, Xiaoyang Gu, Jack H. Lutz & Philippe Moser

  2. Departamento de Informática e Ingeniería de Sistemas, Universidad de Zaragoza, María de Luna 1, 50018, Zaragoza, Spain

    Elvira Mayordomo

Authors
  1. David Doty
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  2. Xiaoyang Gu
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  3. Jack H. Lutz
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  4. Elvira Mayordomo
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  5. Philippe Moser
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Editor information

Editors and Affiliations

  1. Institute of Mathematics, Gdansk University, Wita Stwosza 57, 80 952, Gdansk, Poland

    Joanna Jȩdrzejowicz  & Andrzej Szepietowski  & 

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Doty, D., Gu, X., Lutz, J.H., Mayordomo, E., Moser, P. (2005). Zeta-Dimension. In: Jȩdrzejowicz, J., Szepietowski, A. (eds) Mathematical Foundations of Computer Science 2005. MFCS 2005. Lecture Notes in Computer Science, vol 3618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11549345_25

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  • DOI: https://doi.org/10.1007/11549345_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28702-5

  • Online ISBN: 978-3-540-31867-5

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