On fractal measures and diophantine approximation

We study diophantine properties of a typical point with respect to measures on $\mathbb{R}^n .$ Namely, we identify geometric conditions on a measure μ on $\mathbb{R}^n $ guaranteeing that μ-almost every ${\bf y}\,\in\,\mathbb{R}^n $ 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.