  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.
