Lattice points in large convex planar domains of finite type

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

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