Lattice points in rotated convex domains. (English) Zbl 1359.52022
Let \(B \in \mathbb R^n\) be a compact convex domain with a smooth boundary and containing the origin in the interior. Consider the standard lattice \(\mathbb Z^n\). It is known that the number of lattice points in the dilated domain \(tB\) equals approximately to the volume of the dilated domain, while the reminder is not greater than \(Ct^{n-1}\) asymptotically. In this paper the author proves that for almost every rotation the reminder is of order \(Ct^{n-2+2/(n+1)+\zeta_n}\) with a positive \(\zeta_n\). In the case of the plane (\(n=2\)) this estimate is further extended to general compact convex domains.
Reviewer: Oleg Karpenkov (Liverpool)
MSC:
| 52C07 | Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry) |
| 52C05 | Lattices and convex bodies in \(2\) dimensions (aspects of discrete geometry) |
| 11P21 | Lattice points in specified regions |
| 42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 11L07 | Estimates on exponential sums |
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