Metric Diophantine approximation and 'absolutely friendly' measures

Andrew Pollington; Sanju Velani

doi:10.1007/s00029-005-0007-8

Metric Diophantine approximation and 'absolutely friendly' measures

Andrew Pollington, Sanju Velani

Mathematics

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

Abstract

Let W(¿) denote the set of ¿-well approximable points in \mathbbRd Rdand let K be a compact subset of \mathbbRd Rdwhich supports a measure µ. In this short article, we show that if µ is an ‘absolutely friendly’ measure and a certain µ-volume sum converges then m (W(y) Ç K) = 0.(W()K)=0 The result obtained is in some sense analogous to the convergence part of Khintchine’s classical theorem in the theory of metric Diophantine approximation. The class of absolutely friendly measures is a subclass of the friendly measures introduced in [2] and includes measures supported on self-similar sets satisfying the open set condition. We also obtain an upper bound result for the Hausdorff dimension of W(y) Ç K.W()K

Original language	English
Pages (from-to)	297-307
Number of pages	11
Journal	Selecta Mathematica, New Series
Volume	11
Issue number	2
DOIs	https://doi.org/10.1007/s00029-005-0007-8
Publication status	Published - Dec 2005

Access to Document

10.1007/s00029-005-0007-8

http://arxiv.org/abs/math.NT/0401149

Cite this

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title = "Metric Diophantine approximation and 'absolutely friendly' measures",

abstract = "Let W(¿) denote the set of ¿-well approximable points in \mathbbRd Rdand let K be a compact subset of \mathbbRd Rdwhich supports a measure µ. In this short article, we show that if µ is an {\textquoteleft}absolutely friendly{\textquoteright} measure and a certain µ-volume sum converges then m (W(y) {\c C} K) = 0.(W()K)=0 The result obtained is in some sense analogous to the convergence part of Khintchine{\textquoteright}s classical theorem in the theory of metric Diophantine approximation. The class of absolutely friendly measures is a subclass of the friendly measures introduced in [2] and includes measures supported on self-similar sets satisfying the open set condition. We also obtain an upper bound result for the Hausdorff dimension of W(y) {\c C} K.W()K ",

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AU - Velani, Sanju

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