Optimal stretching for lattice points and eigenvalues

2016/09/05 by Laugesen, Richard, Liu, Shiya · 2 citations
#11P21 #35P15 #FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1609.06172

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. 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. The case p=1 remains open: 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?

Cited by

• The Steklov Spectrum of Spherical Cylinders
• Generalized Hardy's identity for the astroid-type p-circle lattice point problem

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