Kuznecov formulae for fractal measures

Let $(M,g)$ be a compact, connected Riemannian manifold of dimension $n\ge 2$, and let $\{e_j{\}}_{j=0}^{\mathrm{\infty }}$ be an orthonormal basis of Laplace eigenfunctions $-{\mathrm{\Delta }}_g{e}_j={\lambda }_j^2{e}_j$. Given a finite Borel measure $\mu$ on $M$, consider the Kuznecov sum $$N_{\mu }(\lambda ):=\sum _{{\lambda }_j\le \lambda }\mid {\int }_M{e}_{j}d\mu {\mid }^2\mathrm{.}$$ Assume that $\mu$ admits an averaged $s$-density constant $A_{\mu }$ with correlation dimension $s\in (0,n)$. We prove that $$N_{\mu }(\lambda )=(2\pi {)}^{-(n-s)}\mathrm{v}\mathrm{o}\mathrm{l}(B^{n-s})A_{\mu }{\lambda }^{n-s}+o({\lambda }^{n-s})(\lambda \to \mathrm{\infty })\mathrm{.}$$ The averaged $s$-density condition is necessary for such a one-term asymptotic, and in general, the remainder $o({\lambda }^{n-s})$ is sharp in the sense that it cannot be improved uniformly to a power-saving error term. This extends the classical Kuznecov formula of Zelditch for smooth submanifold measures to a broad class of singular and fractal measures.