Nodal intersections of random eigenfunctions against a segment on the
2-dimensional torus

We consider random Gaussian eigenfunctions of the Laplacian on the standard
torus, and investigate the number of nodal intersections against a 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 whether the slope of the straight line is
rational or irrational. Our findings exhibit a close relation between this
problem and the theory of lattice points on circles.