Geodesic Expansions of Off-Diagonal Spectral Functions

Definitions and mathematical statement

Let $M$ be a compact Riemannian surface, let $N_{x,y}(\lambda)$ denote its spectral function, and let $\Gamma_{x,y}$ be the geodesic segments from $x$ to $y$. For a unit-speed segment $\gamma$, write $l(\gamma)$ for its length and $m(\gamma)$ for its Morse index with the intended endpoint convention. The orthogonal Jacobi field in the source has scalar normal component $J$ normalized by $J(0)=0$ and $J'(0)=1$.

For finite $p\geq1$, the usual Besicovitch seminorm on functions of $\lambda\geq0$ is \[ \|h\|_{B^p}=\left(\limsup_{T\to\infty}\frac1T\int_0^T|h(\lambda)|^p\,{\,\mathrm{d}}\lambda\right)^{1/p}. \] The Besicovitch almost-periodic class is obtained by approximation by trigonometric polynomials in this seminorm, identifying functions of seminorm zero; precise completion conventions should be checked against the source.

The question is whether there exists $p\geq1$, chosen for the surface before the points, such that for every $x\in M$ and Riemannian-volume-almost every $y\in M$, \[ \lambda\longmapsto\frac{N_{x,y}(\lambda)}{\sqrt{1+\lambda}} \] has the stated geodesic-sum asymptotic expansion in $B^p$, with frequencies $l(\gamma)$, phases determined by $m(\gamma)$, and amplitudes determined by $l(\gamma)$ and the normalized orthogonal Jacobi field, for $\gamma\in\Gamma_{x,y}$. The exceptional set of $y$ may depend on $x$; the source does not require one common full-measure set for all $x$.

The extraction gives no actual coefficient, phase, or summation formula. It also omits whether $\lambda$ cuts off Laplace eigenvalues or their square roots, the spectral-projector normalization, and the precise meaning of the asymptotic expansion in $B^p$. These omissions prevent writing an exact geodesic series without inventing data.

Short English statement

Seek one Besicovitch exponent for a compact surface so that every fixed basepoint has the proposed off-diagonal geodesic expansion for almost every endpoint. The geometric ingredients and quantifiers are preserved, but the series coefficients and spectral conventions are missing.

Sources

• Local UnsolvedMath registry, permanent ID $30001026$, source code catalogue entry, "Geodesic Expansions of Off-Diagonal Spectral Functions." Registry curation: Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying sources retain their supplied provenance.
• Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2009), Oberwolfach Reports, pp. 2337--2406, DOI https://doi.org/10.4171/owr/2008/42.
• Lapointe, Polterovich and Safarov, supplied title average growth of the spectral function: https://arxiv.org/abs/0803.4171.
• Supplied literature triage, literature_checked_at: 2026-08-21: labeled partially solved; says the Besicovitch almost-periodic expansion is proved for round spheres and flat tori, while existence of a suitable exponent for general surfaces and manifolds remains open. This is supplied dated curation, not independently verified; the statement here retains the surface target rather than introducing an unspecified higher-dimensional normalization.