Lattice points in bodies of revolution II
EntityUAM. Departamento de Matemáticas
10.1002/mana.201800541Mathematische Nachrichten 293.6 (2020): 1074-1083
ISSN0025-584X (print); 1522-2616 (online)
Funded byThe first author is partially supported by the grant MTM2014-56350-P and the MINECO Centro de Excelencia Severo Ochoa Program SEV-2015-0554. The second author has been supported by a “la Caixa”-Severo Ochoa international PhD programme fellowship at the Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM). The second author is indebted to Corentin Perret-Gentil for the fruitful conversations at the MSRI leading to some of the ideas included in this article
ProjectGobierno de España. MTM2014-56350-P
SubjectsExponential sums; Lattice point discrepancy; Van der Corput method; Matemáticas
Note"This is the peer reviewed version of the following article: Mathematische Nachrichten 293.6 (2020): 1074-1083, which has been published in final form at https://doi.org/10.1002/mana.201800541. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited"
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In  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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