  We compute the asymptotic growth rate of the number N(C, R) of closed
geodesics of length less than R in a connected component C of a stratum of
quadratic differentials. We prove that for any 0 < \theta < 1, the number of
closed geodesics of length at most R that spend at least \theta-fraction of
time outside of a compact subset of C is exponentially smaller than N(C, R).
The theorem follows from a lattice counting statement. For points x, y in the
moduli space M of Riemann surfaces, and for 0 < \theta < 1, we find an
upper-bound for the number of geodesic paths of length less than R in C which
connect a point near x to a point near y and spend a \theta-fraction of the
time outside of a compact subset of C.
