Patterson measure and Ubiquity

Maurice Dodson; M.V Melián; D. Pestana; Sanju Velani

Patterson measure and Ubiquity

Maurice Dodson, M.V Melián, D. Pestana, Sanju Velani

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let L be a closed subset of R(k), with Hausdorff dimension delta, which supports a probability measure m for which the m-measure of a ball of radius r and centred at a point in L is comparable to r(delta). By extending the notion of ubiquity from k-dimensional Lebesgue measure to m, a natural lower bound for the Hausdorff dimension of a fairly general class of lim sup subsets of L is obtained. This is applied to Patterson measure supported on the limit set of a convex co-compact group, to obtain the Hausdorff dimension of the set of 'well-approximable' points associated with the limit set. The equivalent geometric result in terms of geodesic excursions on the quotient manifold is also obtained. These results are counterparts of Jarnik's theorem on simultaneous diophantine approximation.

Original language	English
Pages (from-to)	37-60
Number of pages	24
Journal	Annales academiae scientiarum fennicae series a1-Mathematica
Volume	20
Issue number	1
Publication status	Published - 1995

Keywords

DIOPHANTINE APPROXIMATION
HAUSDORFF DIMENSION
SETS

Access to Document

http://www.acadsci.fi/mathematica/Vol20/dodson.pdf

Cite this

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title = "Patterson measure and Ubiquity",

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T1 - Patterson measure and Ubiquity

AU - Dodson, Maurice

AU - Melián, M.V

AU - Pestana, D.

AU - Velani, Sanju

PY - 1995

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N2 - Let L be a closed subset of R(k), with Hausdorff dimension delta, which supports a probability measure m for which the m-measure of a ball of radius r and centred at a point in L is comparable to r(delta). By extending the notion of ubiquity from k-dimensional Lebesgue measure to m, a natural lower bound for the Hausdorff dimension of a fairly general class of lim sup subsets of L is obtained. This is applied to Patterson measure supported on the limit set of a convex co-compact group, to obtain the Hausdorff dimension of the set of 'well-approximable' points associated with the limit set. The equivalent geometric result in terms of geodesic excursions on the quotient manifold is also obtained. These results are counterparts of Jarnik's theorem on simultaneous diophantine approximation.

AB - Let L be a closed subset of R(k), with Hausdorff dimension delta, which supports a probability measure m for which the m-measure of a ball of radius r and centred at a point in L is comparable to r(delta). By extending the notion of ubiquity from k-dimensional Lebesgue measure to m, a natural lower bound for the Hausdorff dimension of a fairly general class of lim sup subsets of L is obtained. This is applied to Patterson measure supported on the limit set of a convex co-compact group, to obtain the Hausdorff dimension of the set of 'well-approximable' points associated with the limit set. The equivalent geometric result in terms of geodesic excursions on the quotient manifold is also obtained. These results are counterparts of Jarnik's theorem on simultaneous diophantine approximation.

KW - DIOPHANTINE APPROXIMATION

KW - HAUSDORFF DIMENSION

KW - SETS

M3 - Article

SN - 0066-1953

VL - 20

SP - 37

EP - 60

JO - Annales academiae scientiarum fennicae series a1-Mathematica

JF - Annales academiae scientiarum fennicae series a1-Mathematica

IS - 1

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