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Higher-rank Bohr sets and multiplicative diophantine approximation
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Chow, Sam and Technau, Niclas (2019) Higher-rank Bohr sets and multiplicative diophantine approximation. Compositio Mathematica, 155 (11). 2214-2233 . doi:10.1112/S0010437X19007589 ISSN 0010-437X.
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Official URL: https://doi.org/10.1112/S0010437X19007589
Abstract
Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Diophantine approximation, Metric spaces, Geometry of numbers , Additive combinatorics | ||||||||
Journal or Publication Title: | Compositio Mathematica | ||||||||
Publisher: | Cambridge University Press | ||||||||
ISSN: | 0010-437X | ||||||||
Official Date: | November 2019 | ||||||||
Dates: |
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Volume: | 155 | ||||||||
Number: | 11 | ||||||||
Page Range: | 2214-2233 | ||||||||
DOI: | 10.1112/S0010437X19007589 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Reuse Statement (publisher, data, author rights): | This article has been published in a revised form in Compositio Mathematica https://doi.org/10.1112/S0010437X19007589. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © The Authors 2019 | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 18 September 2019 | ||||||||
Date of first compliant Open Access: | 18 September 2019 | ||||||||
Related URLs: | |||||||||
Open Access Version: |
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