On the exact Hausdorff dimension of the set of Liouville numbers. II |
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Authors: | L Olsen Dave L Renfro |
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Institution: | (1) Department of Mathematics, University of St. Andrews, St. Andrews, Fife, KY16 9SS, Scotland;(2) ACT, Inc. Iowa City, Iowa 52243, USA |
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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.
This work was done while Dave L. Renfro was at the Department of Mathematics at Central Michigan University. |
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