Abstract.
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.
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Kleinbock, D., Lindenstrauss, E. & Weiss, B. On fractal measures and diophantine approximation. Sel. math., New ser. 10, 479 (2005). https://doi.org/10.1007/s00029-004-0378-2
DOI: https://doi.org/10.1007/s00029-004-0378-2