Arkiv för Matematik
- Ark. Mat.
- Volume 56, Number 1 (2018), 111-145.
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?
Ark. Mat., Volume 56, Number 1 (2018), 111-145.
Received: 23 January 2017
Revised: 8 May 2017
First available in Project Euclid: 19 June 2019
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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