Lattice points in large convex planar domains of finite type

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.

26 pages. This second version gives a slightly better bound than the original version