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Lattice points in rotated convex domains (1303.4137v1)
Published 18 Mar 2013 in math.NT and math.CA
Abstract: 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.