On the Geometry of the Nodal Lines of Eigenfunctions of the Two-Dimensional Torus

Abstract

The width of a convex curve in the plane is the minimal distance between a pair of parallel supporting lines of the curve. In this paper we study the width of nodal lines of eigenfunctions of the Laplacian on the standard flat torus. We prove a variety of results on the width, some having stronger versions assuming a conjecture of Cilleruelo and Granville asserting a uniform bound for the number of lattice points on the circle lying in short arcs.

4 figures. Added some comments about total curvature and other details

References in corpus (1)

• Arithmetical Applications of an Identity for the Vandermonde Determinant

Cited by in corpus (12)

• Phase Singularities in Complex Arithmetic Random Waves
• Planck-scale mass equidistribution of toral Laplace eigenfunctions
• Nodal Lengths in Shrinking Domains for Random Eigenfunctions on ${\mathbb{S}}^2$
• Asymptotic distribution of nodal intersections for arithmetic random waves
• Nodal intersections of random eigenfunctions against a segment on the 2-dimensional torus
• Applications of small scale quantum ergodicity in nodal sets
• CLT for Planck scale mass distribution of toral Laplace eigenfunctions
• On lattices, distinct distances, and the Elekes-Sharir framework
• Small scale equidistribution for a point scatterer on the torus
• Lattice points on small arcs
• Topological Properties of Neumann Domains
• Courant-sharp eigenvalues of a two-dimensional torus