manuscripta mathematica

, Volume 119, Issue 2, pp 217–224

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

Authors

    • Department of MathematicsUniversity of St. Andrews
  • Dave L. Renfro
    • ACT
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 https://static-content.springer.com/image/art%3A10.1007%2Fs00229-005-0604-z/MediaObjects/s00229-005-0604-zflb1.gif denote the set of Liouville numbers. For a dimension function h, we write https://static-content.springer.com/image/art%3A10.1007%2Fs00229-005-0604-z/MediaObjects/s00229-005-0604-zflb2.gif for the h-dimensional Hausdorff measure of https://static-content.springer.com/image/art%3A10.1007%2Fs00229-005-0604-z/MediaObjects/s00229-005-0604-zflb1.gif. In previous work, the exact ``cut-point'' at which the Hausdorff measure https://static-content.springer.com/image/art%3A10.1007%2Fs00229-005-0604-z/MediaObjects/s00229-005-0604-zflb2.gif of https://static-content.springer.com/image/art%3A10.1007%2Fs00229-005-0604-z/MediaObjects/s00229-005-0604-zflb1.gif 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 https://static-content.springer.com/image/art%3A10.1007%2Fs00229-005-0604-z/MediaObjects/s00229-005-0604-zflb2.gif of https://static-content.springer.com/image/art%3A10.1007%2Fs00229-005-0604-z/MediaObjects/s00229-005-0604-zflb1.gif drops from infinity to zero for all dimension functions h. Namely, if h is a dimension function for which the function https://static-content.springer.com/image/art%3A10.1007%2Fs00229-005-0604-z/MediaObjects/s00229-005-0604-zflb3.gif increases faster than any power function near 0, then https://static-content.springer.com/image/art%3A10.1007%2Fs00229-005-0604-z/MediaObjects/s00229-005-0604-zflb4.gif, and if h is a dimension function for which the function https://static-content.springer.com/image/art%3A10.1007%2Fs00229-005-0604-z/MediaObjects/s00229-005-0604-zflb3.gif increases slower than some power function near 0, then https://static-content.springer.com/image/art%3A10.1007%2Fs00229-005-0604-z/MediaObjects/s00229-005-0604-zflb5.gif. This provides a complete characterization of all Hausdorff measures https://static-content.springer.com/image/art%3A10.1007%2Fs00229-005-0604-z/MediaObjects/s00229-005-0604-zflb2.gif of https://static-content.springer.com/image/art%3A10.1007%2Fs00229-005-0604-z/MediaObjects/s00229-005-0604-zflb1.gif without assuming anything about the dimension function h, and answers a question asked by R. D. Mauldin. We also show that if https://static-content.springer.com/image/art%3A10.1007%2Fs00229-005-0604-z/MediaObjects/s00229-005-0604-zflb4.gif then https://static-content.springer.com/image/art%3A10.1007%2Fs00229-005-0604-z/MediaObjects/s00229-005-0604-zflb1.gif does not have σ-finite https://static-content.springer.com/image/art%3A10.1007%2Fs00229-005-0604-z/MediaObjects/s00229-005-0604-zflb6.gif measure. This answers another question asked by R. D. Mauldin.

Copyright information

© Springer-Verlag Berlin Heidelberg 2005