File Download

There are no files associated with this item.

  Links for fulltext
     (May Require Subscription)
Supplementary

Article: On Representation of Integers from Thin Subgroups of SL(2, Z) with Parabolics

TitleOn Representation of Integers from Thin Subgroups of SL(2, Z) with Parabolics
Authors
KeywordsEquidistribution
Discrete Subgroup
Homogeneous Space
Issue Date2020
PublisherOxford University Press. The Journal's web site is located at http://imrn.oxfordjournals.org
Citation
International Mathematics Research Notices, 2020, v. 2020 n. 18, p. 5611-5629 How to Cite?
AbstractLet Λ < SL(2,Z) be a finitely generated, nonelementary Fuchsian group of the 2nd kind, and v,w be two primitive vectors in Z2-0. We consider the set S = {(vγ ,w) R2 : γ ∈Λ}, where ( , ) R2 is the standard inner product in R2. Using Hardy-Littlewood circle method and some infinite co-volume lattice point counting techniques developed by Bourgain, Kontorovich, and Sarnak, together with Gamburd's 5/6 spectral gap, we show that if Λ has parabolic elements, and the critical exponent δ of Λ exceeds 0.998317, then a density-one subset of all admissible integers (i.e., integers passing all local obstructions) are actually in S, with a power savings on the size of the exceptional set (i.e., the set of admissible integers failing to appear in S). This supplements a result of Bourgain-Kontorovich, which proves a density-one statement for the case when Λ is free, finitely generated, has no parabolics, and has critical exponent δ > 0.999950. © The Author(s) 2018.
Persistent Identifierhttp://hdl.handle.net/10722/289435
ISSN
2019 Impact Factor: 1.291
2015 SCImago Journal Rankings: 2.052

 

DC FieldValueLanguage
dc.contributor.authorZhang, X-
dc.date.accessioned2020-10-22T08:12:37Z-
dc.date.available2020-10-22T08:12:37Z-
dc.date.issued2020-
dc.identifier.citationInternational Mathematics Research Notices, 2020, v. 2020 n. 18, p. 5611-5629-
dc.identifier.issn1073-7928-
dc.identifier.urihttp://hdl.handle.net/10722/289435-
dc.description.abstractLet Λ < SL(2,Z) be a finitely generated, nonelementary Fuchsian group of the 2nd kind, and v,w be two primitive vectors in Z2-0. We consider the set S = {(vγ ,w) R2 : γ ∈Λ}, where ( , ) R2 is the standard inner product in R2. Using Hardy-Littlewood circle method and some infinite co-volume lattice point counting techniques developed by Bourgain, Kontorovich, and Sarnak, together with Gamburd's 5/6 spectral gap, we show that if Λ has parabolic elements, and the critical exponent δ of Λ exceeds 0.998317, then a density-one subset of all admissible integers (i.e., integers passing all local obstructions) are actually in S, with a power savings on the size of the exceptional set (i.e., the set of admissible integers failing to appear in S). This supplements a result of Bourgain-Kontorovich, which proves a density-one statement for the case when Λ is free, finitely generated, has no parabolics, and has critical exponent δ > 0.999950. © The Author(s) 2018.-
dc.languageeng-
dc.publisherOxford University Press. The Journal's web site is located at http://imrn.oxfordjournals.org-
dc.relation.ispartofInternational Mathematics Research Notices-
dc.rightsPre-print: Journal Title] ©: [year] [owner as specified on the article] Published by Oxford University Press [on behalf of xxxxxx]. All rights reserved. Pre-print (Once an article is published, preprint notice should be amended to): This is an electronic version of an article published in [include the complete citation information for the final version of the Article as published in the print edition of the Journal.] Post-print: This is a pre-copy-editing, author-produced PDF of an article accepted for publication in [insert journal title] following peer review. The definitive publisher-authenticated version [insert complete citation information here] is available online at: xxxxxxx [insert URL that the author will receive upon publication here].-
dc.subjectEquidistribution-
dc.subjectDiscrete Subgroup-
dc.subjectHomogeneous Space-
dc.titleOn Representation of Integers from Thin Subgroups of SL(2, Z) with Parabolics-
dc.typeArticle-
dc.identifier.emailZhang, X: xz27@hku.hk-
dc.identifier.authorityZhang, X=rp02608-
dc.description.naturelink_to_subscribed_fulltext-
dc.identifier.doi10.1093/imrn/rny177-
dc.identifier.scopuseid_2-s2.0-85080856424-
dc.identifier.hkuros317219-
dc.identifier.volume2020-
dc.identifier.issue18-
dc.identifier.spage5611-
dc.identifier.epage5629-
dc.publisher.placeUnited Kingdom-

Export via OAI-PMH Interface in XML Formats


OR


Export to Other Non-XML Formats