Zeta-Dimension

(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)

Abstract

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

Dimζ(A) = inf{s | ζA(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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adleman, L.: Toward a mathematical theory of self-assembly. Technical report, USC (January 2000)Google Scholar
  2. 2.
    Apostol, T.M.: Introduction to Analytic Number Theory. In: Undergraduate Texts in Mathematics, Springer, Heidelberg (1976)Google Scholar
  3. 3.
    Apostol, T.M.: Modular Functions and Dirichlet Series in Number Theory. Graduate Texts in Mathematics 41 (1976)Google Scholar
  4. 4.
    Athreya, K.B., Hitchcock, J.M., Lutz, J.H., Mayordomo, E.: Effective strong dimension, algorithmic information, and computational complexity. SIAM Journal on Computing (to appear); Preliminary version appeared. In: Proceedings of the 21st International Symposium on Theoretical Aspects of Computer Science, pp. 632–643 (2004)Google Scholar
  5. 5.
    Barlow, M.T., Taylor, S.J.: Fractional dimension of sets in discrete spaces. Journal of Physics A 22, 2621–2626 (1989)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Barlow, M.T., Taylor, S.J.: Defining fractal subsets of ℤd. Proc. London Math. Soc. 64, 125–152 (1992)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bedford, T., Fisher, A.: Analogues of the lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc. 64, 95–124 (1992)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cahen, E.: Sur la fonction ζ(s) de Riemann et sur des fonctions analogues. Annales de l’École Normale supérieure 11(3), S. 85 (1894)Google Scholar
  9. 9.
    Conner, W.M.: The dimension of a formal language. Information and Control 29, 1–10 (1975)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dai, J.J., Lathrop, J.I., Lutz, J.H., Mayordomo, E.: Finite-state dimension. Theoretical Computer Science 310, 1–33 (2004)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    deLuca, A.: On the entropy of a formal language. In: Brakhage, H. (ed.) Automata Theory and Formal Languages. Proc. 2nd GI Conference. LNCS, vol. (33), pp. 103–109. Springer, Heidelberg (1975)Google Scholar
  12. 12.
    Dirichlet, L.: Über den satz: das jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz keinen gemeinschaftlichen Factor sind, unendlichen viele Primzahlen enthalt. Mathematische Abhandlungen,1837. Bd. 1 313–342 (1889)Google Scholar
  13. 13.
    Eilenberg, S.: Automata, Languages, and Machines, vol. A. Academic Press, London (1974)MATHGoogle Scholar
  14. 14.
    Euler, L.: Variae observationes circa series infinitas. Commentarii Academiae Scientiarum Imperialis Petropolitanae 9, 160–188 (1737)Google Scholar
  15. 15.
    Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. Wiley, Chichester (2003)MATHCrossRefGoogle Scholar
  16. 16.
    Furstenberg, H.: Intersections of Cantor sets and transversality of semigroups. In: Problems in analysis. Symposium Salomon Bochner (1969)Google Scholar
  17. 17.
    Hansel, G., Perrin, D., Simon, I.: Compression and entropy. In: Proceedings of the 9th Annual Symposium on Theoretical Aspects of Computer Science, pp. 515–528 (1992)Google Scholar
  18. 18.
    Hardy, G., Wright, E.: An Introduction to the Theory of Numbers, 5th edn. Clarendon Press, Oxford (1979)MATHGoogle Scholar
  19. 19.
    Hausdorff, F.: Dimension und äusseres Mass. Mathematische Annalen 79, 157–179 (1919)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Hitchcock, J.M.: Effective fractal dimension: foundations and applications. PhD thesis, Iowa State University (2003)Google Scholar
  21. 21.
    Hueter, I., Peres, Y.: Self-affine carpets on the square lattice. Comb. Probab., Computing 6, 197–204 (1997)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Knopp, K.: Über die Abszisse der Grenzgeraden einer Dirichletschen Reihe. Sitzungsberichte der Berliner Mathematischen Gesellschaft (1910)Google Scholar
  23. 23.
    Knopp, K.: Theory and Application of Infinite Series. Dover Publications, New York (1990) ( First published in German in 1921 and in English in 1928)Google Scholar
  24. 24.
    Kuich, W.: On the entropy of context-free languages. Information and Control 16(2), 173–200 (1970)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Li, M., Vitányi, P.M.B.: An Introduction to Kolmogorov Complexity and its Applications, 2nd edn. Springer, Berlin (1997)MATHGoogle Scholar
  26. 26.
    Lutz, J.H.: Dimension in complexity classes. SIAM Journal on Computing 32, 1236–1259 (2003)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Lutz, J.H.: The dimensions of individual strings and sequences. Information and Computation 187, 49–79 (2003)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Lutz, J.H.: Effective fractal dimensions. Mathematical Logic Quarterly 51, 62–72 (2005)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    McKenzie, P., Wagner, K.: The complexity of membership problems for circuits over sets of natural numbers. In: Proceedings of the Twentieth Annual Symposium on Theoretical Aspects of Computer Science, pp. 571–582 (2003)Google Scholar
  30. 30.
    Olsen, L.: Distribution of digits in integers: fractal dimension and zeta functions. Acta Arith. 105(3), 253–277 (2002)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Olsen, L.: Distribution of digits in integers: Besicovitch-Eggleston subsets of ℕ. Journal of London Mathematical Society 67(3), 561–579 (2003)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Riemann, B.: Über die Anzahl der Primzahlen unter einer gegebener Grösse. In: Monatsber. Akad. Berlin, pp. 671–680 (1859)Google Scholar
  33. 33.
    Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares (extended abstract). In: STOC, pp. 459–468 (2000)Google Scholar
  34. 34.
    Shannon, C.E.: A mathematical theory of communication. Bell System Technical Journal 27, 379–423, 623–656 (1948)MATHMathSciNetGoogle Scholar
  35. 35.
    Staiger, L.: Kolmogorov complexity and Hausdorff dimension. Information and Computation 103, 159–194 (1993)MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Stewart, I.: Four encounters with Sierpinski’s gasket. The Mathematical Intelligencer 17(1), 52–64 (1995)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Strichartz, R.S.: Fractals in the large. Canadian Journal of Mathematics 50(3), 638–657 (1998)MATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Sullivan, D.: Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Mathematica 153, 259–277 (1984)MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Tricot, C.: Two definitions of fractional dimension. Mathematical Proceedings of the Cambridge Philosophical Society 91, 57–74 (1982)MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Willson, S.J.: Growth rates and fractional dimensions in cellular automata. Physica D 10, 69–74 (1984)CrossRefMathSciNetGoogle Scholar
  41. 41.
    Willson, S.J.: The equality of fractional dimensions for certain cellular automata. Physica D 24, 179–189 (1987)MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Wirsing, E.: Bemerkung zu der arbeit über vollkommene zahlen. Mathematische Annalen 137, 316–318 (1959)MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Zvonkin, A.K., Levin, L.A.: The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms. Russian Mathematical Surveys 25, 83–124 (1970)MATHCrossRefMathSciNetGoogle Scholar

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

Personalised recommendations