On fractal measures and diophantine approximation

• Dmitry Kleinbock ^[1] ; Elon Lindenstrauss ^[2] ; Barak Weiss ^[3]
  1. [1] Brandeis University

     Brandeis University

     City of Waltham, Estados Unidos

     
  2. [2] Stanford University

     Stanford University

     Estados Unidos

     
  3. [3] Ben-Gurion University of the Negev

     Ben-Gurion University of the Negev

     Israel

     

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 10, Nº. 4, 2005, págs. 479-523
• Idioma: inglés
• DOI: 10.1007/s00029-004-0378-2
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• Resumen

  • We study diophantine properties of a typical point with respect to measures on Rn. Namely, we identify geometric conditions on a measure μ on Rn guaranteeing that μ-almost every y∈Rn is not very well multiplicatively approximable by rationals. Measures satisfying our conditions are called ‘friendly’. Examples include smooth measures on nondegenerate manifolds; thus this paper generalizes the main result of [KM]. Another class of examples is given by measures supported on self-similar sets satisfying the open set condition, as well as their products and pushforwards by certain smooth maps.