manuscripta mathematica

, Volume 119, Issue 2, pp 217–224

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

Article

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.

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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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