Event description
The set of real numbers which are badly approximable by rationals admits an exhaustion by sets Bad(c), whose dimension converges to 1 as c goes to zero. D. Hensley computed the asymptotic for the dimension up to the first order in c, via an analogous estimate for the set of real numbers whose continued fraction has all entries uniformly bounded. We consider diophantine approximations by parabolic fixed points of any non-uniform lattice in PSL(2,R) and a geometric notion of c-badly approximable points. We compute the dimension of the set of such points up to the first order in c, via the thermodynamic method of Ruelle and Bowen. Geometric good approximations are related to a notion of bounded partial quotients for the Bowen-Series expansion. This gives a family of Cantor sets and associated quasi-compact transfer operators, with simple and positive maximal eigenvalue. Perturbative analysis of spectra applies.