Lattice points in large convex planar domains of finite type

TL;DR

This paper establishes that for most rotations of a smooth, convex planar domain of finite type, the error term in counting lattice points grows no faster than a specific sublinear rate, uniformly across rotations.

Contribution

It proves a uniform bound on the lattice point remainder for almost all rotations of convex domains of finite type, extending previous results to a broader class of shapes.

Findings

Remainder term is of order $O(t^{2/3- ext{positive constant}})$ for almost every rotation.

The result holds uniformly over all rotations, independent of the specific domain.

The bound improves understanding of lattice point distribution in convex domains.

Abstract

Let $\mathcal{B}$ be a compact convex planar domain with smooth boundary of finite type and $\mathcal{B}_\theta$ its rotation by an angle $\theta$. We prove that for almost every $\theta\in[0, 2\pi]$ the remainder $P_{\mathcal{B}_\theta}(t)$ is of order $O_{\theta}(t^{2/3-\zeta})$ with a positive number $\zeta$ independent of the domain.