Metric Diophantine approximation and ‘absolutely friendly’ measures

• Andrew Pollington ^[1] ; Sanju L. Velani ^[2]
  1. [1] Brigham Young University

     Brigham Young University

     Estados Unidos

     
  2. [2] University of York

     University of York

     Reino Unido

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 11, Nº. 2, 2005, págs. 297-307
• Idioma: inglés
• DOI: 10.1007/s00029-005-0007-8
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• Resumen

  • Let W(ψ) denote the set of ψ-well approximable points in Rd and let K be a compact subset of Rd which 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 μ(W(ψ)∩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(ψ)∩K.