manuscripta mathematica

, Volume 119, Issue 2, pp 217–224

On the exact Hausdorff dimension of the set of Liouville numbers. II

Article

DOI: 10.1007/s00229-005-0604-z

Cite this article as:
Olsen, L. & Renfro, D. manuscripta math. (2006) 119: 217. doi:10.1007/s00229-005-0604-z

Abstract

Let Open image in new window denote the set of Liouville numbers. For a dimension function h, we write Open image in new window for the h-dimensional Hausdorff measure of Open image in new window. In previous work, the exact ``cut-point'' at which the Hausdorff measure Open image in new window of Open image in new window drops from infinity to zero has been located for various classes of dimension functions h satisfying certain rather restrictive growth conditions. In the paper, we locate the exact ``cut-point'' at which the Hausdorff measure Open image in new window of Open image in new window drops from infinity to zero for all dimension functions h. Namely, if h is a dimension function for which the function Open image in new window increases faster than any power function near 0, then Open image in new window, and if h is a dimension function for which the function Open image in new window increases slower than some power function near 0, then Open image in new window. This provides a complete characterization of all Hausdorff measures Open image in new window of Open image in new window without assuming anything about the dimension function h, and answers a question asked by R. D. Mauldin. We also show that if Open image in new window then Open image in new window does not have σ-finite Open image in new window measure. This answers another question asked by R. D. Mauldin.

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of St. AndrewsFifeScotland
  2. 2.ACTInc. Iowa CityUSA

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