Variations on a Theorem of Birman and Series

Suppose that $\mathrm{\Sigma }$ is a hyperbolic surface and $f:\mathbb R_+\to\mathbb R_+$ a monotonic function. We study the closure in the projective tangent bundle $PT\mathrm{\Sigma }$ of the set of all geodesics $\gamma$ satisfying $I(\gamma ,\gamma )\le f({\mathrm{\ell }}_{\mathrm{\Sigma }}(\gamma ))$. For instance we prove that if $f$ is unbounded and sublinear then this set has Hausdorff dimension strictly bounded between 1 and 3.