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

- First Online:

- Received:

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

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## Abstract

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