Research output: Contribution to journal › Article

**Patterson measure and Ubiquity.** / Dodson, Maurice; Melián, M.V; Pestana, D.; Velani, Sanju.

Research output: Contribution to journal › Article

Dodson, M, Melián, MV, Pestana, D & Velani, S 1995, 'Patterson measure and Ubiquity', *Annales academiae scientiarum fennicae series a1-Mathematica*, vol. 20, no. 1, pp. 37-60.

Dodson, M., Melián, M. V., Pestana, D., & Velani, S. (1995). Patterson measure and Ubiquity. *Annales academiae scientiarum fennicae series a1-Mathematica*, *20*(1), 37-60.

Dodson M, Melián MV, Pestana D, Velani S. Patterson measure and Ubiquity. Annales academiae scientiarum fennicae series a1-Mathematica. 1995;20(1):37-60.

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title = "Patterson measure and Ubiquity",

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.",

keywords = "DIOPHANTINE APPROXIMATION, HAUSDORFF DIMENSION, SETS",

author = "Maurice Dodson and M.V Meli{\'a}n and D. Pestana and Sanju Velani",

year = "1995",

language = "English",

volume = "20",

pages = "37--60",

journal = "Annales academiae scientiarum fennicae series a1-Mathematica",

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TY - JOUR

T1 - Patterson measure and Ubiquity

AU - Dodson, Maurice

AU - Melián, M.V

AU - Pestana, D.

AU - Velani, Sanju

PY - 1995

Y1 - 1995

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

VL - 20

SP - 37

EP - 60

JO - Annales academiae scientiarum fennicae series a1-Mathematica

JF - Annales academiae scientiarum fennicae series a1-Mathematica

SN - 0066-1953

IS - 1

ER -