Lattice points in bodies of revolution II

Fernando Chamizo and Carlos Pastor

Mathematische Nachrichten, 2020, vol. 293, issue 6, 1074-1083

Abstract: In [3] 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.

Date: 2020
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https://doi.org/10.1002/mana.201800541

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