Arkiv för Matematik

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

Optimal stretching for lattice points and eigenvalues

Richard S. Laugesen and Shiya Liu

Full-text: Open access

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
Received: 23 January 2017
Revised: 8 May 2017
First available in Project Euclid: 19 June 2019

Permanent link to this document
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

Subjects
Primary: 35P15: Estimation of eigenvalues, upper and lower bounds
Secondary: 11P21: Lattice points in specified regions 52C05: Lattices and convex bodies in $2$ dimensions [See also 11H06, 11H31, 11P21]

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

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


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