Tubular Neighborhoods of Nodal Sets and Diophantine Approximation

TL;DR

This paper investigates the volume of tubular neighborhoods around nodal sets of Laplacian eigenfunctions on real analytic manifolds and explores their application in Diophantine approximation, revealing new bounds and analogies.

Contribution

It provides new upper and lower bounds on tubular neighborhood volumes and connects nodal set approximation with Diophantine approximation on manifolds.

Findings

Established bounds on tubular neighborhood volumes.

Linked nodal set approximation to rational number approximation.

Enhanced understanding of eigenfunction nodal sets in geometric analysis.

Abstract

We give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the Laplacian on a real analytic closed Riemannian manifold M. As an application we consider the question of approximating points on M by nodal sets, and explore analogy with approximation by rational numbers.