On the nodal sets of toral eigenfunctions

We study the nodal sets of eigenfunctions of the Laplacian on the standard d-dimensional flat torus. The question we address is: Can a fixed hypersurface lie on the nodal sets of eigenfunctions with arbitrarily large eigenvalue? In dimension two, we show that this happens only for segments of closed geodesics. In higher dimensions, certain cylindrical sets do lie on nodal sets corresponding to arbitrarily large eigenvalues. Our main result is that this cannot happen for hypersurfaces with nonzero Gauss-Kronecker curvature. In dimension two, the result follows from a uniform lower bound for the L²-norm of the restriction of eigenfunctions to the curve, proved in an earlier paper (Bourgain and Rudnick in C. R. Math. 347(21-22):1249-1253, 2009). In high dimensions we currently do not have this bound. Instead, we make use of the real-analytic nature of the flat torus to study variations on this bound for restrictions of eigenfunctions to suitable submanifolds in the complex domain. In all of our results, we need an arithmetic ingredient concerning the cluster structure of lattice points on the sphere. We also present an independent proof for the two-dimensional case relying on the "abc-theorem" in function fields.