TY - JOUR
T1 - On the exact Hausdorff dimension of the set of Liouville numbers
AU - Olsen, Lars Ole Ronnow
PY - 2005/2
Y1 - 2005/2
N2 - Let L denote the set of Liouville numbers. For a dimension function h, we write H-h(L) for the h-dimensional Hausdorff measure of L. In this paper we locate the exact "cut-point" at which the Hausdorff measure of L drops from infinity to zero. Namely, if h is a dimension function that increases faster than any power function near 0, then H-h(L) = infinity, and if h is a dimension function that increases slower than some power function near 0, then H-h(L) = 0. This answers a question asked by R. D. Mauldin.
AB - Let L denote the set of Liouville numbers. For a dimension function h, we write H-h(L) for the h-dimensional Hausdorff measure of L. In this paper we locate the exact "cut-point" at which the Hausdorff measure of L drops from infinity to zero. Namely, if h is a dimension function that increases faster than any power function near 0, then H-h(L) = infinity, and if h is a dimension function that increases slower than some power function near 0, then H-h(L) = 0. This answers a question asked by R. D. Mauldin.
UR - http://www.scopus.com/inward/record.url?scp=14644402476&partnerID=8YFLogxK
U2 - 10.1007/s00229-004-0528-z
DO - 10.1007/s00229-004-0528-z
M3 - Article
SN - 0025-2611
VL - 116
SP - 157
EP - 172
JO - Manuscripta Mathematica
JF - Manuscripta Mathematica
ER -