Equidistribution of toral eigenfunctions along hypersurfaces

TL;DR

This paper establishes quantum variance estimates for toral eigenfunctions and demonstrates their equidistribution along hypersurfaces with nonvanishing curvature, confirming conjectures and extending understanding of eigenfunction behavior on flat tori.

Contribution

It introduces new quantum variance bounds and proves equidistribution of eigenfunctions along hypersurfaces, verifying Bourgain-Rudnick's conjecture in any dimension.

Findings

Existence of a density one subsequence of eigenfunctions equidistributing along hypersurfaces.

Verification of Bourgain-Rudnick's conjecture on $L^2$-restriction estimates.

Equidistribution results against measures with Fourier dimension > d-2.

Abstract

We prove a new quantum variance estimate for toral eigenfunctions. As an application, we show that, given any orthonormal basis of toral eigenfunctions and any smooth embedded hypersurface with nonvanishing principal curvatures, there exists a density one subsequence of eigenfunctions that equidistribute along the hypersurface. This is an analogue of the Quantum Ergodic Restriction theorems in the case of the flat torus, which in particular verifies the Bourgain-Rudnick's conjecture on $L^2$-restriction estimates for a density one subsequence of eigenfunctions in any dimension. Using our quantum variance estimates, we also obtain equidistribution of eigenfunctions against measures whose supports have Fourier dimension larger than $d-2$. In the end, we also describe a few quantitative results specific to dimension $2$.