  Given a hyperbolic surface $\S$, a classic result of Birman and Series states
that for each $K$, all complete geodesics with at most $K$ self-intersections
can only pass through a certain nowhere dense, Hausdorff dimension 1 subset of
$\S$. We define a self-intersection function for each complete geodesic, which
bounds the number of self-intersections in finite length subarcs. We then
extend the Birman-Series result to sets of complete geodesics with certain
bounds on their self-intersection functions. In fact, we get the same
conclusion as the Birman-Series result for sets of complete geodesics whose
self-intersection functions are in $o(l^2)$, where $l$ measures arclength.
