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 -