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Bohr sets and multiplicative diophantine approximation
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Chow, Sam (2018) Bohr sets and multiplicative diophantine approximation. Duke Mathematical Journal, 167 (9). pp. 1623-1642. doi:10.1215/00127094-2018-0001 ISSN 0012-7094.
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Official URL: https://doi.org/10.1215/00127094-2018-0001
Abstract
In two dimensions, Gallagher’s theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fiber version of Gallagher’s theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes, and Velani. The idea is to find large generalized arithmetic progressions within inhomogeneous Bohr sets, extending a construction given by Tao. This precise structure enables us to verify the hypotheses of the Duffin–Schaeffer theorem for the problem at hand, via the geometry of numbers.
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, Additive combinatorics, Geometry of numbers | ||||||
Journal or Publication Title: | Duke Mathematical Journal | ||||||
Publisher: | Duke University Press | ||||||
ISSN: | 0012-7094 | ||||||
Official Date: | 23 May 2018 | ||||||
Dates: |
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Volume: | 167 | ||||||
Number: | 9 | ||||||
Page Range: | pp. 1623-1642 | ||||||
DOI: | 10.1215/00127094-2018-0001 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||
Date of first compliant deposit: | 19 September 2019 | ||||||
Date of first compliant Open Access: | 19 September 2019 | ||||||
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