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

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.