On quantum limits on flat tori

We classify all weak $*$ limits of squares of normalized eigenfunctions of the Laplacian on two-dimensional flat tori (we call these limits quantum limits). We also obtain several results about such limits in dimensions three and higher. Many of the results are a consequence of a geometric lemma which describes a property of simplices of codimension one in $\mathbb {R}^n$ whose vertices are lattice points on spheres. The lemma follows from the finiteness of the number of solutions of a system of two Pell equations. A consequence of the lemma is a generalization of the result of B. Connes. We also indicate a proof (communicated to us by J. Bourgain) of the absolute continuity of the quantum limits on a flat torus in any dimension. We generalize a two-dimensional result of Zygmund to three dimensions; we discuss various possible generalizations of that result to higher dimensions and the relation to $L^p$ norms of the densities of quantum limits and their Fourier series.