Metric Diophantine approximation and ‘ absolutely friendly ’ measures

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 note, 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 Khintchines 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.