Badly approximable numbers and vectors in Cantor-like sets

We show that a large class of Cantor-like sets of ${{\mathbb{R}}}^{d}, d \geq 1$, contains uncountably many badly approximable numbers, respectively badly approximable vectors, when $d\geq 2$. An analogous result is also proved for subsets of ${\mathbb{R}}^d$ arising in the study of geodesic flows corresponding to $(d+1)$-dimensional manifolds of constant negative curvature and finite volume, generalizing the set of badly approximable numbers in ${\mathbb{R}}$. Furthermore, we describe a condition on sets, which is fulfilled by a large class, ensuring a large intersection with these Cantor-like sets.