Nodal intersections for random waves against a segment on the
3-dimensional torus

We consider random Gaussian eigenfunctions of the Laplacian on the
three-dimensional flat torus, and investigate the number of nodal intersections
against a straight line segment. The expected intersection number, against any
smooth curve, is universally proportional to the length of the reference curve,
times the wavenumber, independent of the geometry. We found an upper bound for
the nodal intersections variance, depending on the arithmetic properties of the
straight line. The considerations made establish a close relation between this
problem and the theory of lattice points on spheres.