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

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of St. Andrews, St. Andrews, Fife, KY16 9SS, Scotland

    L. Olsen

  2. ACT, Inc. Iowa City, Iowa, 52243, USA

    Dave L. Renfro

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  1. L. Olsen
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  2. Dave L. Renfro
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Correspondence to L. Olsen.

Additional information

This work was done while Dave L. Renfro was at the Department of Mathematics at Central Michigan University.

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Olsen, L., Renfro, D. On the exact Hausdorff dimension of the set of Liouville numbers. II. manuscripta math. 119, 217–224 (2006). https://doi.org/10.1007/s00229-005-0604-z

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  • DOI: https://doi.org/10.1007/s00229-005-0604-z

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