旋转体中的格点II
Lattice points in bodies of revolution II
摘要：
在以前的文章中，它表明，当一个三维光滑凸体具有围绕坐标轴的旋转对称性，可以找到更好的边界格点差异比已知的更一般的凸体。然而，要做到这一点，必须假设母线的三阶导数为非零条件。在这篇文章中，我们放弃了这个条件，表明上述界限适用于更广泛的旋转体族，其中包括具有解析边界的旋转体。在我们的方法的一个新颖之处是，除了通常的分析方法，它需要研究的相位上的傅里叶变换侧的泰勒系数的丢番图性质。
英文摘要：
In a previous article it was shown that when a three-dimensional smooth convex body has rotational symmetry around a coordinate axis one can find better bounds for the lattice point discrepancy than what is known for more general convex bodies. To accomplish this, however, it was necessary to assume a non-vanishing condition on the third derivative of the generatrix. In this article we drop this condition, showing that the aforementioned bound holds for a wider family of revolution bodies, which includes those with analytic boundary. A novelty in our approach is that, besides the usual analytic methods, it requires studying some Diophantine properties of the Taylor coefficients of the phase on the Fourier transform side.
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