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# Arkiv för Matematik

## Volume 56 (2018)

### Number 1

### Optimal stretching for lattice points and eigenvalues

Pages: 111 – 145

DOI: http://dx.doi.org/10.4310/ARKIV.2018.v56.n1.a8

#### Authors

#### 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?

#### 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.

Received 23 January 2017

Received revised 8 May 2017