Lattice points in bodies of revolution II

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‐vanishin...

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Bibliographic Details
Published in:Mathematische Nachrichten 2020-06, Vol.293 (6), p.1074-1083
Main Authors: Chamizo, Fernando, Pastor, Carlos
Format: Article
Language:English
Subjects:
Bodies of revolution
exponential sums
Fourier transforms
lattice point discrepancy
Mathematical analysis
Phase transitions
van der Corput method
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Summary: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.
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.201800541