Badly approximable vectors on fractals (Q814135)

The aim of the present paper is to describe a large class of closed subsets \(C\) of \(\mathbb{R}^n\) which contain badly approximable vectors since there is no analogous description for \(n>1\) but only very few examples. In particular the paper is focused on presenting that the intersection of \(C\) (describing in terms of geometric properties of measures they support) with the set of badly approximable vectors has the same Hausdorff dimension as \(C\). Under this study the results are new even in the case \(n=1\). As applications to the above consideration the paper gives examples which include self-similar sets such as Cantor's ternary sets or attractors for irreducible systems of similarities satisfying Hutchinson's open set condition.