On numbers badly approximable by dyadic rationals

We consider a problem originating both from circle coverings and badly approximable numbers in the case of dyadic diophantine approximation. For the unit circle S we give an elementary proof that the set {x is an element of S : 2(n)x >= c (mod 1) n >= 0} is a fractal set whose Hausdorff dimension depends continuously on c and is constant on intervals which form a set of Lebesgue measure 1. Hence it has a fractal graph. We completely characterize the intervals where the dimension remains unchanged. As a consequence we can describe the graph of c bar right arrow dim(H) {x is an element of [0, 1] : x - m/2(n) < c/2(n) (mod 1) finitely often}