Gauss lattices and complex continued fractions
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Publication:2071463
DOI10.4310/PAMQ.2021.V17.N5.A6zbMATH Open1486.11091arXiv2101.05480OpenAlexW4210282138WikidataQ114020790 ScholiaQ114020790MaRDI QIDQ2071463
Author name not available (Why is that?)
Publication date: 28 January 2022
Published in: (Search for Journal in Brave)
Abstract: Our aim is to find a complex continued fraction algorithm finding all the best Diophantine approximations to a complex number. Using the sequence of minimal vectors in a two dimensional lattice over Gaussian integers, we obtain an algorithm defined on a submanifold of the space of unimodular two dimensional Gauss lattices. This submanifold is transverse to the diagonal flow. Thanks to the correspondence between minimal vectors and best Diophantine approximations, the algorithm finds all the best approximations to a complex number. A byproduct of the algorithm is the best constant for the complex version of Dirichlet Theorem about approximations of complex numbers by quotients of Gaussian integers.
Full work available at URL: https://arxiv.org/abs/2101.05480
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