Optimal stretching for lattice points and eigenvalues

Abstract

We aim to maximize the number of first-quadrant lattice points in a convex domain with respect to reciprocal stretching in the coordinate directions. The optimal domain is shown to be asymptotically balanced, meaning that the stretch factor approaches 1 as the "radius" approaches infinity. In particular, the result implies that among all p-ellipses (or Lam\'e curves), the p-circle encloses the most first-quadrant lattice points as the radius approaches infinity. The case p=2 corresponds to minimization of high eigenvalues of the Dirichlet Laplacian on rectangles, and so our work generalizes a result of Antunes and Freitas. Similarly, we generalize a Neumann eigenvalue maximization result of van den Berg, Bucur and Gittins. The case p=1 remains open: which right triangles in the first quadrant with two sides along the axes will enclose the most lattice points, as the area tends to infinity?