TY - JOUR

T1 - On the exact Hausdorff dimension of the set of Liouville numbers. II

AU - Olsen, Lars Ole Ronnow

AU - Renfro, Dave L.

PY - 2006/2

Y1 - 2006/2

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 previous work, the exact " cutpoint" at which the Hausdorff measure H h( L) of L 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 H h( L) of L drops from infinity to zero for all dimension functions h. Namely, if h is a dimension function for which the function r -> inf(0 < s <= r) r (h(s))(/s) s increases faster than any power function near 0, then H-h( L)=infinity, and if h is a dimension function for which the function r -> inf (0 < s <= r) r (h(s))(/s) s increases slower than some power function near 0, then H-h( L) = 0. This provides a complete characterization of all Hausdorff measures H h( L) of L without assuming anything about the dimension function h, and answers a question asked by R. D. Mauldin. We also show that if H-h( L)=infinity, then L does not have sigma- finite H-h measure. This answers another 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 previous work, the exact " cutpoint" at which the Hausdorff measure H h( L) of L 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 H h( L) of L drops from infinity to zero for all dimension functions h. Namely, if h is a dimension function for which the function r -> inf(0 < s <= r) r (h(s))(/s) s increases faster than any power function near 0, then H-h( L)=infinity, and if h is a dimension function for which the function r -> inf (0 < s <= r) r (h(s))(/s) s increases slower than some power function near 0, then H-h( L) = 0. This provides a complete characterization of all Hausdorff measures H h( L) of L without assuming anything about the dimension function h, and answers a question asked by R. D. Mauldin. We also show that if H-h( L)=infinity, then L does not have sigma- finite H-h measure. This answers another question asked by R. D. Mauldin.

KW - APPROXIMATION

UR - http://www.scopus.com/inward/record.url?scp=32144449291&partnerID=8YFLogxK

U2 - 10.1007/s00229-005-0604-z

DO - 10.1007/s00229-005-0604-z

M3 - Article

SN - 0025-2611

VL - 119

SP - 217

EP - 224

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

ER -