Arkiv för Matematik
Volume 56 (2018)
Optimal stretching for lattice points and eigenvalues
Pages: 111 – 145
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?
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.
Received 23 January 2017
Received revised 8 May 2017
Accepted 9 August 2017
Published 30 April 2018