Optimal stretching for lattice points and eigenvalues

Research output: Contribution to journalArticle

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<p<∞. 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<p<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?.

Original languageEnglish (US)
Pages (from-to)111-145
Number of pages35
JournalArkiv for Matematik
Volume56
Issue number1
DOIs
StatePublished - Jan 1 2018

Keywords

  • Dirichlet eigenvalues
  • Lamé curve
  • Laplacian
  • Lattice points
  • Neumann eigenvalues
  • P-ellipse
  • Planar convex domain
  • Spectral optimization

ASJC Scopus subject areas

  • Mathematics(all)

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