It seems to me that the results in this important paper of Kahn-Markovich imply the following fact. Let $M^3$ be any closed hyperbolic 3-manifold. For every $\epsilon > 0$ there is a natural number $R(\epsilon)$ such that for every $R>R(\epsilon)>0$ there is a closed geodesic whose lenght is between $R-\epsilon$ and $R+\epsilon$.

That is, the more $R$ grows, the more the spectrum of all geodesics having length more than $R$ becomes uniformly crowded, without holes. Am I right that this is a consequence of their results? If so, is there a more direct way to prove this? Does this property generalize to closed hyperbolic manifolds with arbitrary dimension $n$?