# Weyl asymptotic in towers

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?