Jean Lagace

Uniform estimates on lattice point counts and toral eigenvalues

Counting lattice points inside a disk or a ball is a problem dating back to Gauß in the early 19th century, and corresponds to counting the eigenvalues of the Laplacian on a flat torus below some energy level. It has been long known that the number is asymptotically proportional to the volume of the ball divided by the determinant of the lattice, up to a comparatively small error. This error, however, depends strongly on the lattice, which is a significant hindrance in analysing optimisation questions such as "What is the largest determinant that a lattice with exactly k points in the unit ball can have". I will discuss this question, it's links with spectral shape optimisation and further directions in which we could now go.