Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-29T19:02:48.006Z Has data issue: false hasContentIssue false
  • English
  • Français

The sum of Rademacher functions and Hausdorff dimension

Published online by Cambridge University Press:  24 October 2008

Tian-You Hu  and
Ka-Sing Lau
Tian-You Hu
Affiliation:
Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.
Ka-Sing Lau
Affiliation:
Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

For 0 < α < 1, let for 0 ≤ x < 1, where is the sequence of Rademacher functions. We give a class of fα so that their graphs have Hausdorff dimension 2 − α. The result is closely related to the corresponding unsolved question for the Weierstrass functions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 1 , July 1990 , pp. 97 - 103
Copyright
Copyright © Cambridge Philosophical Society 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Beyer, W. A.. Hausdorff dimension of level sets of some Rademacher series. Pacific J. Math. 12 (1962), 35–46.CrossRefGoogle Scholar
[2]Erdös, P.. On the smoothness properties of a family of Bernoulli convolutions. Amer. J. Math. 62 (1940), 180–186.CrossRefGoogle Scholar
[3]Falconer, K. J.. The Geometry of Fractal Sets (Cambridge University Press, 1985).CrossRefGoogle Scholar
[4]Garsia, A. M.. Arithmetic properties of Bernoulli convolutions. Trans. Amer. Math. Soc. 102 (1962), 409–132.CrossRefGoogle Scholar
[5]Hardy, G. H. and Littlewood, J. E.. Some properties of fractional integrals. Math. Z. 27 (1928), 565–606.CrossRefGoogle Scholar
[6]Jessen, B. and Wintner, A.. Distribution functions and the Riemann zeta function. Trans. Amer. Math. Soc. 38 (1935), 48–88.CrossRefGoogle Scholar
[7]Kahane, J. P. and Salem, R.. Sur la convolution d'une infinité de distributions de Bernoulli. Colloq. Math. 6 (1958), 193–202.CrossRefGoogle Scholar
[8]Mauldin, R. D. and Williams, S. C.. On the Hausdorff dimension of some graphs. Trans. Amer. Math. Soc. 298 (1986), 793–803.CrossRefGoogle Scholar
[9]Salem, R.. Algebraic Numbers and Fourier Analysis (Heath, 1963).Google Scholar
[10]Wintner, A.. On convergent Poisson convolutions. Amer. J. Math. 57 (1935), 827–838.CrossRefGoogle Scholar