On numbers badly approximable by dyadic rationals

Abstract

The thesis takes as starting point diophantine approximation with focus on the area of badly approximable numbers. For the special kind of rationals, the q-adic rationals, we consider two types of approimations models, a one-sided and a two-sided model, and the sets of badly approximable numbers they give rise to. We prove with elementary methods that the Hausdorff dimension of these two sets depends continuously on a defining parameter, is constant Lebesgue almost every and self similar. Hence they are fractal sets. Moreover, we give the complete description of the intervals where the dimension remains unchanged. The methods and techniques in the proofs uses ideas form symbolic dynamics, combinatorics on words and the beta-shift.