Lattice point congurations and arithmetic

This talk is based on collaborations with Dmitry Beliaev, Jacques Benatar, and Maurizia Rossi. The nodal (zero) sets of Laplace eigenfunctions remain stationary during membrane vibrations. Their study dates back to the 17th Century. These eigenfunctions, called `arithmetic waves', have several physical applications e.g. telecommunications, the study of ocean waves, sound and music. In this talk, we will discuss the surprising connections with analytic number theory and discrete geometry. These include the distribution and correlations of lattice points on circles and spheres, and their Riesz s-energy. Moreover, we will see how the spectral measure of the arithmetic wave allows us to detect the limiting distribution of the lattice points.