Lattice points in rotated convex domains

TL;DR

This paper investigates the distribution of lattice points in rotated convex domains, establishing almost sure bounds on the remainder term in higher dimensions and extending planar estimates to general convex shapes.

Contribution

It proves almost sure bounds for the lattice point remainder in rotated convex domains in higher dimensions and extends planar estimates to all compact convex domains.

Findings

Remainder of lattice point problem is of order $O_{ heta}(t^{d-2+2/(d+1)- zeta_d})$ for almost every rotation.

Extension of planar convex domain estimates to general convex domains.

Results hold for convex domains with smooth boundary of finite type.

Abstract

If $\mathcal{B}\subset \mathbb{R}^d$ ($d\geqslant 2$) is a compact convex domain with a smooth boundary of finite type, we prove that for almost every rotation $\theta\in SO(d)$ the remainder of the lattice point problem, $P_{\theta \mathcal{B}}(t)$, is of order $O_{\theta}(t^{d-2+2/(d+1)-\zeta_d})$ with a positive number $\zeta_d$. Furthermore we extend the estimate of the above type, in the planar case, to general compact convex domains.