Effective Dimensions and Relative Frequencies

  • Xiaoyang Gu
  • Jack H. Lutz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5028)

Abstract

Consider the problem of calculating the fractal dimension of a set X consisting of all infinite sequences S over a finite alphabet Σ that satisfy some given condition P on the asymptotic frequencies with which various symbols from Σ appear in S. Solutions to this problem are known in cases where

(i) the fractal dimension is classical (Hausdorff or packing dimension), or

(ii) the fractal dimension is effective (even finite-state) and the condition Pcompletely specifies an empirical distribution π over Σ, i.e., a limiting frequency of occurrence for every symbol in Σ.

In this paper we show how to calculate the finite-state dimension (equivalently, the finite-state compressibility) of such a set X when the condition P only imposes partial constraints on the limiting frequencies of symbols. Our results automatically extend to less restrictive effective fractal dimensions (e.g., polynomial-time, computable, and constructive dimensions), and they have the classical results (i) as immediate corollaries. Our methods are nevertheless elementary and, in most cases, simpler than those by which the classical results were obtained.

Keywords

effective fractal dimensions empirical frequencies finite-state dimension randomness saturated sets 

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References

  1. 1.
    Athreya, K.B., Hitchcock, J.M., Lutz, J.H., Mayordomo, E.: Effective strong dimension, algorithmic information, and computational complexity. SIAM Journal on Computing 37, 671–705 (2007)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Barreira, L., Saussol, B., Schmeling, J.: Distribution of frequencies of digits via multifractal analyais. Journal of Number Theory 97(2), 410–438 (2002)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Besicovitch, A.S.: On the sum of digits of real numbers represented in the dyadic system. Mathematische Annalen 110, 321–330 (1934)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cajar, H.: Billingsley dimension in probability spaces. Lecture notes in mathematics, vol. 892 (1981)Google Scholar
  5. 5.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory. John Wiley & Sons, Inc., New York (1991)MATHGoogle Scholar
  6. 6.
    Dai, J.J., Lathrop, J.I., Lutz, J.H., Mayordomo, E.: Finite-state dimension. Theoretical Computer Science 310, 1–33 (2004)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Eggleston, H.: The fractional dimension of a set defined by decimal properties. Quarterly Journal of Mathematics 20, 31–36 (1949)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Falconer, K.: The Geometry of Fractal Sets. Cambridge University Press, Cambridge (1985)MATHGoogle Scholar
  9. 9.
    Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. Wiley, Chichester (2003)MATHGoogle Scholar
  10. 10.
    Good, I.J.: The fractional dimensional theory of continued fractions. In: Proceedings of the Cambridge Philosophical Society, vol. 37, pp. 199–228 (1941)Google Scholar
  11. 11.
    Gu, X.: A note on dimensions of polynomial size circuits. Theoretical Computer Science 359(1-3), 176–187 (2006)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gu, X., Lutz, J.H., Moser, P.: Dimensions of Copeland-Erdős sequences. Information and Computation 205(9), 1317–1333 (2007)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hausdorff, F.: Dimension und äusseres Mass. Mathematische Annalen 79, 157–179 (1919)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Hitchcock, J.M.: Fractal dimension and logarithmic loss unpredictability. Theoretical Computer Science 304(1–3), 431–441 (2003)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Huffman, D.A.: A method for the construction of minimum redundancy codes. In: Proc. IRE, vol. 40, pp. 1098–1101 (1952)Google Scholar
  16. 16.
    Kelly, J.: A new interpretation of information rate. Bell Systems Technical Journal 35, 917–926 (1956)Google Scholar
  17. 17.
    Lutz, J.H.: Gales and the constructive dimension of individual sequences. In: Proceedings of the 27th International Colloquium on Automata, Languages, and Programming, pp. 902–913 (2000); Revised as [19]Google Scholar
  18. 18.
    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)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lutz, J.H.: The dimensions of individual strings and sequences. Information and Computation 187, 49–79 (2003); Preliminary version appeared as [17]MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    McMullen, C.T.: Hausdorff dimension of general Sierpinski carpets. Nagoya Mathematical Journal 96, 1–9 (1984)MATHMathSciNetGoogle Scholar
  21. 21.
    Olsen, L.: Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. Journal de Mathématiques Pures et Appliquées. Neuvième Série 82(12), 1591–1649 (2003)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Olsen, L.: Applications of multifractal divergence points to some sets of d-tuples of numbers defined by their n-adic expansion. Bulletin des Sciences Mathématiques 128(4), 265–289 (2004)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Olsen, L.: Applications of divergence points to local dimension functions of subsets of ℝd. Proceedings of the Edinburgh Mathematical Society 48, 213–218 (2005)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Olsen, L.: Multifractal analysis of divergence points of the deformed measure theoretical Birkhoff averages. III. Aequationes Mathematicae 71(1-2), 29–53 (2006)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Olsen, L., Winter, S.: Multifractal analysis of divergence points of the deformed measure theoretical Birkhoff averages II (preprint, 2001)Google Scholar
  26. 26.
    Olsen, L., Winter, S.: Normal and non-normal points of self-similar sets and divergence points of self-similar measures. Journal of the London Mathematical Society (Second Series) 67(1), 103–122 (2003)MATHMathSciNetGoogle Scholar
  27. 27.
    Ryabko, B., Suzuki, J., Topsoe, F.: Hausdorff dimension as a new dimension in source coding and predicting. In: 1999 IEEE Information Theory Workshop, pp. 66–68 (1999)Google Scholar
  28. 28.
    Shannon, C.E.: A mathematical theory of communication. Bell System Technical Journal 27, 379–423, 623–656 (1948)MathSciNetMATHGoogle Scholar
  29. 29.
    Sullivan, D.: Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Mathematica 153, 259–277 (1984)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Tricot, C.: Two definitions of fractional dimension. Mathematical Proceedings of the Cambridge Philosophical Society 91, 57–74 (1982)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Volkmann, B.: Über Hausdorffsche Dimensionen von Mengen, die durch Zifferneigenschaften charakterisiert sind. VI. Mathematische Zeitschrift 68, 439–449 (1958)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

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

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