Metric Diophantine approximation and ‘absolutely friendly’ measures

Abstract.

Let W(ψ) denote the set of ψ-well approximable points in $\mathbb{R}^{d}$ and let K be a compact subset of $\mathbb{R}^{d}$ 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 $\mu\,(W(\psi)\,\cap\,K)\,=\,0.$ 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 $W(\psi)\,\cap\,K.$