Lattice points in rotated convex domains

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 $θ\in SO(d)$ the remainder of the lattice point problem, $P_{θ\mathcal{B}}(t)$, is of order $O_θ(t^{d-2+2/(d+1)-ζ_d})$ with a positive number $ζ_d$. Furthermore we extend the estimate of the above type, in the planar case, to general compact convex domains.

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