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

TL;DR

This paper investigates the geometric properties of nodal lines of Laplacian eigenfunctions on the 2D torus, focusing on their width and related lattice point distribution conjectures.

Contribution

It provides new results on the width of nodal lines and explores implications of a conjecture regarding lattice points on circles.

Findings

Results on bounds for the width of nodal lines

Connections to lattice point distribution conjectures

Stronger results assuming the Cilleruelo-Granville conjecture

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.