Optimal stretching for lattice points and eigenvalues
  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? 
Richard Laugesen, Shiya Liu
math.MG math.NT math.SP