Let $M$ be a compact Riemannian manifold. The Weyl law gives the asymptotic of the counting function $N(T)$ of Laplace eigenvalues $|\lambda|\le T$ as $T\to\infty$. Now suppose you are given a tower of connected coverings $M_{j+1}\to M_j\to M$ converging to the universal covering, i.e., we have $\bigcap_j\pi_1(M_j)=\{1\}$, where the intersection takes place in $\pi_1(M)$. Then you get 2-parameter counting function $N(j,T)$. What is known about the combined asymptotics, as $j$ and $T$ tend to infinity independently?



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