Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-30T21:40:43.672Z Has data issue: false hasContentIssue false
  • English
  • Français

On the exact rate of convergence of frequencies of digits and local dimensions of multinomial measures

Published online by Cambridge University Press:  07 August 2015

L. OLSEN
L. OLSEN*
Affiliation:
Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the Hausdorff dimensions of certain sets of non-normal numbers defined in terms of the exact rate of convergence of digits in their N-adic expansions. As an application of our results we analyse the rate of convergence of local dimensions of multinomial measures.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 3 , November 2015 , pp. 387 - 403
Copyright
Copyright © Cambridge Philosophical Society 2015 

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

[CM] Cawley, R. and Mauldin, R. D. Multifractal decomposition of Moran fractals. Adv. Math. 92 (1992), 196–236.Google Scholar
[Ed] Edgar, G. Integral, Probability and Fractal Measures (Springer–Verlag, New York, 1998).Google Scholar
[Fa1] Falconer, K. J. Fractal Geometry–Mathematical Foundations and Applications (John Wiley and Sons, 1990).Google Scholar
[Fa2] Falconer, K. J. Techniques in Fractal Geometry (John Wiley and Sons, Ltd., Chichester, 1997).Google Scholar
[FS] Fan, A. H. and Schmeling, J. On fast Birkhoff averaging. Math. Proc. Camb. Phil. Soc. 135 (2003), 443–467.Google Scholar
[JP] Jordan, T. and Pollicott, M. Multifractal analysis and the variance of Gibbs measures. J. London Math. Soc. 76 (2007), 57–72.Google Scholar
[MR] Morán, M. and Rey, J.-M.. Singularity of self-similar measures with respect to Hausdorff measures. Trans. Amer. Math. Soc. 350 (1998), 2297–2310.Google Scholar
[Ol1] Olsen, L. Slow and fast convergence to local dimensions of self-similar measures. Math. Nachr. 266 (2004), 68–80.Google Scholar
[Ol2] Olsen, L. On the exact rate of convergence of frequencies of digits in self-similar sets. Indag. Math. 17 (2006), 85–102.Google Scholar