Kuznecov formulae for fractal measures

Abstract

Let $(M,g)$ be a compact, connected Riemannian manifold of dimension $n\ge 2$, and let $\{e_j{\}}_{j=0}^{\infty }$ be an orthonormal basis of Laplace eigenfunctions $-{\Delta }_g{e}_j={\lambda }_j^2{e}_j$. Given a finite Borel measure $\mu$ on $M$, consider the Kuznecov sum \[ N_μ(λ):=\sum_{λ_j\le λ}\Bigl|\int_M e_j\,dμ\Bigr|^2. \] Assume that $\mu$ admits an averaged $s$-density constant $A_{\mu }$ with correlation dimension $s\in (0,n)$. We prove that \[N_μ(λ)= (2π)^{-(n-s)}\,{\rm vol}(B^{\,n-s})\,A_μ\,λ^{n-s}+ o(λ^{n-s})\qquad (λ\to\infty). \] 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.

arXiv comment22 pages, minor edits