Nodal intersections and Lp restriction theorems on the torus

TL;DR

This paper investigates the intersection properties of eigenfunction nodal lines on the torus with a fixed curve, proposing bounds related to lattice points and restriction norms, connecting geometric and arithmetic aspects.

Contribution

It introduces a new lower bound for nodal intersections on the torus based on lattice point estimates, linking geometric properties to arithmetic conjectures.

Findings

Lower bound differs from wave number by an arithmetic quantity

Connection between lattice points in arcs and nodal intersections

Reduction of the problem to restriction norm bounds

Abstract

We study the number of intersections of the nodal lines of an eigenfunction of the Laplacian on the standard torus with a fixed reference curve, that is, the number of zeros of the eigenfunction restricted to the curve. An upper bound is the wave number k. When the curve has nowhere zero curvature, we conjecture that, up to a constant multiple, this should also be the correct a lower bound. We give a lower bound which differs from this by an arithmetic quantity, given in terms of the maximal number of lattice points in arcs of size square root of the wave number k on a circle of radius k. According to a conjecture of Cilleruelo and Granville, this quantity is bounded in which case we recover our conjecture. To get at the lower bound, we reduce the problem to giving a lower bound for the L1 norm of the restriction of the eigenfunction to the curve, and then to an upper bound for the L4 restriction norm.