Lattice points in rotated convex domains

Abstract

If $\mathcal{B}\subset {\mathbb{R}}^d$ ($d⩾2$) 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.