Optimal stretching for lattice points under convex curves

Sinan Ariturk, Richard S. Laugesen

Research output: Contribution to journalArticle

Abstract

Suppose we count the positive integer lattice points beneath a convex decreasing curve in the first quadrant having equal intercepts. Then stretch in the coordinate directions so as to preserve the area under the curve, and again count lattice points. Which choice of stretch factor will maximize the lattice point count? We show the optimal stretch factor approaches 1 as the area approaches infinity. In particular, when 0 < p < 1, among p-ellipses |sx|p + |s-1y|p = rp with s > 0, the one enclosing the most first-quadrant lattice points approaches a p-circle (s = 1) as r →∞. The case p = 2 was established by Antunes and Freitas, with generalization to 1 < p < ∞ by Laugesen and Liu. The behavior in the borderline case p = 1 (lattice points in right triangles) is quite different, as shown recently by Marshall and Steinerberger.

Original languageEnglish (US)
Pages (from-to)91-114
Number of pages24
JournalPortugaliae Mathematica
Volume74
Issue number2
DOIs
StatePublished - Jan 1 2017

Keywords

  • Lattice points
  • P-ellipse
  • Planar domain

ASJC Scopus subject areas

  • Mathematics(all)

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