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 Lamé curves), the $p$-circle encloses the most first-quadrant lattice points as the radius approaches infinity, for $1\lt p \lt \infty$.

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. Further, Ariturk and Laugesen recently handled $0 \lt p \lt 1$ by building on our results here.

The case $p=1$ remains open, and is closely related to minimizing energy levels of harmonic oscillators: 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?

Keywords

lattice points, planar convex domain, $p$-ellipse, Lamé curve, spectral optimization, Laplacian, Dirichlet eigenvalues, Neumann eigenvalues

2010 Mathematics Subject Classification

Primary 35P15. Secondary 11P21, 52C05.

Full Text (PDF format)

Received 23 January 2017

Received revised 8 May 2017

Accepted 9 August 2017

Published 30 April 2018