Fishman, Lior and Simmons, David orcid.org/0000-0002-9136-6635 (2016) Variations on Dirichlet's theorem. Journal of Number Theory. pp. 11-22. ISSN 0022-314X
Abstract
We give a necessary and sufficient condition for the following property of an integer d∈N and a pair (a,A)∈R2: There exist κ>0 and Q0∈N such that for all x∈Rd and Q≥Q0, there exists p/q∈Qd such that 1≤q≤Q and x-p/q≤κq-aQ-A. This generalizes Dirichlet's theorem, which states that this property holds (with κ=Q0=1) when a=1 and A=1/d. We also analyze the set of exceptions in those cases where the statement does not hold, showing that they form a comeager set. This is also true if Rd is replaced by an appropriate "Diophantine space", such as a nonsingular rational quadratic hypersurface which contains rational points. Finally, in the case d=1 we describe the set of exceptions in terms of classical Diophantine conditions.
Metadata
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Copyright, Publisher and Additional Information: | © 2015 Elsevier Inc. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. | ||||
Keywords: | Diophantine approximation, Dirichlet's theorem | ||||
Dates: |
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Institution: | The University of York | ||||
Academic Units: | The University of York > Faculty of Sciences (York) > Mathematics (York) | ||||
Funding Information: |
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Depositing User: | Pure (York) | ||||
Date Deposited: | 16 Mar 2017 12:40 | ||||
Last Modified: | 06 Dec 2023 11:44 | ||||
Published Version: | https://doi.org/10.1016/j.jnt.2015.10.011 | ||||
Status: | Published | ||||
Refereed: | Yes | ||||
Identification Number: | https://doi.org/10.1016/j.jnt.2015.10.011 | ||||
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