  The set of real numbers which are badly approximable by rationals admits an
exhaustion by sets Bad($\epsilon$), whose dimension converges to 1 as
$\epsilon$ goes to zero. D. Hensley computed the asymptotic for the dimension
up to the first order in $\epsilon$, 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 $\epsilon$-badly
approximable points. We compute the dimension of the set of such points up to
the first order in $\epsilon$, 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.
