## Arkiv för Matematik

• Ark. Mat.
• Volume 56, Number 1 (2018), 111-145.

### 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 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?

#### Article information

Source
Ark. Mat., Volume 56, Number 1 (2018), 111-145.

Dates
Revised: 8 May 2017
First available in Project Euclid: 19 June 2019

https://projecteuclid.org/euclid.afm/1560967223

Digital Object Identifier
doi:10.4310/ARKIV.2018.v56.n1.a8

Mathematical Reviews number (MathSciNet)
MR3800462

Zentralblatt MATH identifier
1390.35206

#### Citation

Laugesen, Richard S.; Liu, Shiya. Optimal stretching for lattice points and eigenvalues. Ark. Mat. 56 (2018), no. 1, 111--145. doi:10.4310/ARKIV.2018.v56.n1.a8. https://projecteuclid.org/euclid.afm/1560967223