Lattice points in stretched model domains of finite type in $\mathbb{R}^d$

TL;DR

This paper investigates how to optimally stretch certain convex domains in higher dimensions to maximize lattice point counts, revealing that the optimal shape becomes asymptotically balanced, with implications for spectral geometry.

Contribution

It introduces a new optimal stretching problem for convex domains with boundary points of vanishing Gaussian curvature in  dimensions, providing asymptotic results and bounds.

Findings

Optimal domain is asymptotically balanced.

Two-term bounds for lattice counting are established.

Explicit Fourier transform estimates are derived.

Abstract

We study an optimal stretching problem for certain convex domain in $\mathbb{R}^d$ ($d\geq 3$) whose boundary has points of vanishing Gaussian curvature. We prove that the optimal domain which contains the most positive (or least nonnegative) lattice points is asymptotically balanced. This type of problem has its origin in the "eigenvalue optimization among rectangles" problem in spectral geometry. Our proof relies on two-term bounds for lattice counting for general convex domains in $\mathbb{R}^d$ and an explicit estimate of the Fourier transform of the characteristic function associated with the specific domain under consideration.