Revista Matemática Iberoamericana

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Volume 31, Issue 2, 2015, pp. 411–438
DOI: 10.4171/RMI/839

Published online: 2015-07-16

Lattice points in rotated convex domains

Jingwei Guo[1]

(1) University of Scinece and Technology of China, Hefei, Anhui, China

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.

Keywords: Lattice points, convex domains, Fourier transform, Van der Corput’s method

Guo Jingwei: Lattice points in rotated convex domains. Rev. Mat. Iberoam. 31 (2015), 411-438. doi: 10.4171/RMI/839