Weyl Law Improvement for Products of Spheres

TL;DR

This paper improves the classical Weyl Law for product manifolds, specifically for products of spheres, by refining the error term from big-O to little-o, and explores related lattice point distribution and conjectures.

Contribution

It demonstrates that the Weyl Law error term can be improved for product manifolds, especially spheres, and connects this to lattice point distribution and additive combinatorics.

Findings

Error term improved to o(λ^{d-1}) for product manifolds

Reduced problem to weighted lattice point distribution

Formulated a conjecture related to sum-product phenomena

Abstract

The classical Weyl Law says that if $N_M(\lambda)$ denotes the number of eigenvalues of the Laplace operator on a $d$-dimensional compact manifold $M$ without a boundary that are less than or equal to $\lambda$, then $$ N_M(\lambda)=c\lambda^d+O(\lambda^{d-1}).$$ In this paper, we show Duistermaat and Guillemin's result allows us to replace the $O(\lambda^{d-1})$ error with $o(\lambda^{d-1})$ if $M$ is a product manifold. We quantify this bound in the case of Cartesian product of spheres by reducing the problem to the study of the distribution of weighted integer lattice points in Euclidean space and formulate a conjecture in the general case reminiscent of the sum-product phenomenon in additive combinatorics.