BADLY APPROXIMABLE NUMBERS AND VECTORS IN CANTOR-LIKE SETS

Abstract

Abstract. We show that a large class of Cantor-like sets of R d,d ≥ 1, contains uncountably many badly approximable numbers, respectively badly approximable vectors, when d ≥ 2. An analogous result is also proved for subsets of 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 R. Furthermore, we describe a condition on sets, which is fulfilled by a large class, ensuring a large intersection with these Cantor-like sets. 1