Skip to main content
SpringerLink
Log in

Hausdorff dimension of sets arising in number theory

  • Conference paper
  • First Online:
Number Theory

Part of the book series: Lecture Notes in Mathematics ((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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. K.I. Babenko. On a problem of Gauss; Dokl. Akad. Nauk SSSR 238 (1978), 1021–1024 Soviet Math. Dok. 19 (1978), 136–140.

    MathSciNet  MATH  Google Scholar 

  2. E. Best. On sets of fractional dimension, III; Proc. London Math. Soc. (2) 47 (1942), 436–454.

    Article  MathSciNet  MATH  Google Scholar 

  3. P. Billingsley. Ergodic theory and information. Wiley, New York 1965.

    MATH  Google Scholar 

  4. D.W. Boyd. The residual set dimension of the Apollonian packing; Mathematika 20 (1973), 170–174.

    Article  MathSciNet  MATH  Google Scholar 

  5. R.T. Bumby. Hausdorff dimensions of Cantor sets; J. reine angew. Math. 331 (1982), 192–206.

    MathSciNet  MATH  Google Scholar 

  6. T.W. Cusick. Continuants with bounded digits, I; Mathematika 24 (1977), 166–127. ____, II; ibid. 25 (1978), 107–108.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Gillis. Note on a theorem of Myrberg; Proc. Camb. Phil. Soc. 33 (1937), 419–424.

    Article  MATH  Google Scholar 

  8. I.J. Good. The fractional dimension theory of continued fractions; Proc. Camb. Phil. Soc. 37 (1941), 199–228.

    Article  MathSciNet  MATH  Google Scholar 

  9. G.H. Hardy. A course of pure mathematics (10th edition). Cambridge University Press, Cambridge 1952.

    MATH  Google Scholar 

  10. F. Hausdorff. Dimension und äusseres Mass; Math. Ann. 79 (1919), 157–179.

    Article  MathSciNet  MATH  Google Scholar 

  11. B. Mandelbrot. Fractals: form, chance, and dimension. Freeman, San Francisco 1977.

    MATH  Google Scholar 

  12. C.A. Rogers. Hausdorff measures. Cambridge University Press, Cambridge 1970.

    MATH  Google Scholar 

  13. A.L. Schmidt. Ergodic theory for complex continued fractions; Monatsh. Math. 93 (1982), 39–62.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  15. H. Wegmann. Das Hausdorff-Mass von Cantormengen; Math. Ann. 193 (1971), 7–20.

    Article  MathSciNet  Google Scholar 

  16. A. Weil. L'intégration dans les groupes topologique et ses applications. Hermann, Paris 1951.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Rutgers University, USA

    Richard T. Bumby

Authors
  1. Richard T. Bumby
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

David V. Chudnovsky Gregory V. Chudnovsky Harvey Cohn Melvyn B. Nathanson

Rights and permissions

Reprints and permissions

Copyright information

© 1985 Springer-Verlag

About this paper

Cite this paper

Bumby, R.T. (1985). Hausdorff dimension of sets arising in number theory. In: Chudnovsky, D.V., Chudnovsky, G.V., Cohn, H., Nathanson, M.B. (eds) Number Theory. Lecture Notes in Mathematics, vol 1135. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074599

Download citation

  • DOI: https://doi.org/10.1007/BFb0074599

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-15649-9

  • Online ISBN: 978-3-540-39535-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation