The hyperbolic geometry of Markov’s theorem on Diophantine approximation and quadratic forms

dc.contributor.authorSpringborn, Boris
dc.date.accessioned2019-01-31T09:59:38Z
dc.date.available2019-01-31T09:59:38Z
dc.date.issued2017
dc.descriptionDieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.de
dc.descriptionThis publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.en
dc.description.abstractMarkov’s theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to present a new proof of Markov’s theorem using hyperbolic geometry. The main ingredients are a dictionary to translate between hyperbolic geometry and algebra/number theory, and some very basic tools borrowed from modern geometric Teichmüller theory. Simple closed geodesics and ideal triangulations of the modular torus play an important role, and so do the problems: How far can a straight line crossing a triangle stay away from the vertices? How far can it stay away from all vertices of the tessellation generated by this triangle? Definite binary quadratic forms are briefly discussed in the last section.en
dc.description.sponsorshipDFG, SFB/TR 109, Discretization in Geometry and Dynamicsen
dc.identifier.eissn2309-4672
dc.identifier.issn0013-8584
dc.identifier.urihttps://depositonce.tu-berlin.de//handle/11303/9051
dc.identifier.urihttp://dx.doi.org/10.14279/depositonce-8152
dc.language.isoenen
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subject.ddc510 Mathematikde
dc.subject.othermodular torusen
dc.subject.othersimple closed geodesicen
dc.subject.otherMarkov equationen
dc.subject.otherFord circlesen
dc.subject.otherFarey tessellationen
dc.titleThe hyperbolic geometry of Markov’s theorem on Diophantine approximation and quadratic formsen
dc.typeArticleen
dc.type.versionpublishedVersionen
dcterms.bibliographicCitation.doi10.4171/LEM/63-3/4-5en
dcterms.bibliographicCitation.issue3-4en
dcterms.bibliographicCitation.journaltitleL' enseignement mathématiqueen
dcterms.bibliographicCitation.originalpublishernameEuropean Mathematical Societyen
dcterms.bibliographicCitation.originalpublisherplaceZürichen
dcterms.bibliographicCitation.pageend373en
dcterms.bibliographicCitation.pagestart333en
dcterms.bibliographicCitation.volume63en
tub.accessrights.dnbdomainen
tub.affiliationFak. 2 Mathematik und Naturwissenschaften>Inst. Mathematikde
tub.affiliation.facultyFak. 2 Mathematik und Naturwissenschaftende
tub.affiliation.instituteInst. Mathematikde
tub.publisher.universityorinstitutionTechnische Universität Berlinen
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