Optimal stretching for lattice points under convex curves

TL;DR

This paper investigates how to optimally stretch convex curves in the first quadrant to maximize lattice points beneath them, showing the optimal stretch approaches no stretching as the area increases, with implications for eigenvalue problems.

Contribution

It extends the understanding of lattice point maximization under convex curves, especially for p-ellipses with 0<p<1, and connects to eigenvalue minimization conjectures in higher dimensions.

Findings

Optimal stretch factor approaches 1 as area tends to infinity.

For p-ellipses with 0<p<1, the maximal lattice points approach the p-circle.

Supports conjecture that minimal eigenvalue boxes approach cubes in high dimensions.

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 $|sx|^p+|s^{-1}y|^p=r^p$ with $s>0$, the one enclosing the most first-quadrant lattice points approaches a $p$-circle ($s=1$) as $r \to \infty$. The case $p=2$ was established by Antunes and Freitas, with generalization to $1<p<\infty$ by Laugesen and Liu. The case $p=1$ remains open, where the question is: 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? Our results for $p<1$ lend support to the conjecture that in all dimensions, the rectangular box of given volume that minimizes the $n$-th eigenvalue of the Dirichlet Laplacian will approach a cube as $n \to \infty$. This conjecture remains open in dimensions four and higher.