Resumen de Metric Diophantine approximation and ‘absolutely friendly’ measures

Andrew Pollington, Sanju L. Velani

• 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.