Lattice points in stretched finite type domains

TL;DR

This paper investigates how to optimally stretch convex, symmetric domains in multi-dimensional space to maximize lattice points, showing that the best configurations tend to be balanced as the domain size grows.

Contribution

It introduces an optimal stretching problem for convex domains with smooth, finite type boundaries and proves that optimal solutions are asymptotically balanced.

Findings

Optimal domains contain the most lattice points are asymptotically balanced.

The study extends lattice point problems to stretched convex domains.

Symmetry and smoothness conditions are crucial for the results.

Abstract

We study an optimal stretching problem, which is a variant lattice point problem, for convex domains in $\mathbb{R}^d$ ($d\geq 2$) with smooth boundary of finite type that are symmetric with respect to each coordinate hyperplane/axis. We prove that optimal domains which contain the most positive (or least nonnegative) lattice points are asymptotically balanced.