Hausdorff dimension of sets arising in number theory

  • Richard T. Bumby
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1135)

Abstract

The sets considered here have their roots in arithmetic, but the theoretical tools introduced to compute their Hausdorff dimensions should have broader interest and application. In particular, the relation of the Hausdorff dimension to the spectral radius of the subdivision operator provides a means of eliminating ad hoc estimates, thereby sharpening the calculations. The use of monotonicity to allow inequalities of functions to be tested by finite numerical calculations does not seem to have a place in the numerical analysis arsenal. It bears further study. The "spectral analysis" given by equation (4) illustrate a self-duality which seems to be present also for the circle-packing example. This is likely to be an important structure.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K.I. Babenko. On a problem of Gauss; Dokl. Akad. Nauk SSSR 238 (1978), 1021–1024 Soviet Math. Dok. 19 (1978), 136–140.MathSciNetMATHGoogle Scholar
  2. 2.
    E. Best. On sets of fractional dimension, III; Proc. London Math. Soc. (2) 47 (1942), 436–454.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    P. Billingsley. Ergodic theory and information. Wiley, New York 1965.MATHGoogle Scholar
  4. 4.
    D.W. Boyd. The residual set dimension of the Apollonian packing; Mathematika 20 (1973), 170–174.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    R.T. Bumby. Hausdorff dimensions of Cantor sets; J. reine angew. Math. 331 (1982), 192–206.MathSciNetMATHGoogle Scholar
  6. 6.
    T.W. Cusick. Continuants with bounded digits, I; Mathematika 24 (1977), 166–127. ____, II; ibid. 25 (1978), 107–108.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    J. Gillis. Note on a theorem of Myrberg; Proc. Camb. Phil. Soc. 33 (1937), 419–424.CrossRefMATHGoogle Scholar
  8. 8.
    I.J. Good. The fractional dimension theory of continued fractions; Proc. Camb. Phil. Soc. 37 (1941), 199–228.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    G.H. Hardy. A course of pure mathematics (10th edition). Cambridge University Press, Cambridge 1952.MATHGoogle Scholar
  10. 10.
    F. Hausdorff. Dimension und äusseres Mass; Math. Ann. 79 (1919), 157–179.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    B. Mandelbrot. Fractals: form, chance, and dimension. Freeman, San Francisco 1977.MATHGoogle Scholar
  12. 12.
    C.A. Rogers. Hausdorff measures. Cambridge University Press, Cambridge 1970.MATHGoogle Scholar
  13. 13.
    A.L. Schmidt. Ergodic theory for complex continued fractions; Monatsh. Math. 93 (1982), 39–62.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    B. Volkmann. Über Hausdorffsche Dimensionen von Mengen die durch Zifferneigenschaften charakterisiert sind I; Math. Z. 58 (1953), 284–287. ___, II; ibid. 59 (1953), 247–254. ___, III; ibid. 59 (1953), 259–270. ___, IV; ibid. 59 (1954), 425–433. ____, V; ibid. 65 (1956), 389–413.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    H. Wegmann. Das Hausdorff-Mass von Cantormengen; Math. Ann. 193 (1971), 7–20.MathSciNetCrossRefGoogle Scholar
  16. 16.
    A. Weil. L'intégration dans les groupes topologique et ses applications. Hermann, Paris 1951.Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Richard T. Bumby
    • 1
  1. 1.Rutgers UniversityUSA

Personalised recommendations