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Algorithms and Computation in Mathematics 26 Oleg N. Karpenkov Geometry of Continued Fractions Second Edition Algorithms and Computation in Mathematics Volume 26 Series Editors William J. Cook, Mathematics, University of Waterloo, Waterloo, ON, Canada David Eisenbud, Berkeley, CA, USA Bernhard Korte, Research Institute for Discrete Mathematics, University of Bonn, Bonn, Germany László Lovász, Számítógéptudományi Tanszék, Eötvös Loránd University, Budapest, Hungary Michael F. Singer, Mathematics, North Carolina State University, Raleigh, NC, USA Bernd Sturmfels, Berkeley, CA, USA BiancaViray,DepartmentofMathematics,UniversityofWashington,Seattle,WA, USA Avi Wigderson, School of Mathematics, Institute for Advanced Study, Princeton University, Princeton, NJ, USA Günter M. Ziegler , Institut für Mathematik, Freie Universität Berlin, Berlin, Germany With this forward-thinking series Springer recognizes that the prevailing trend in mathematical research towards algorithmic and constructive processes is one of long-term importance. This series is intended to further the development of computational and algorithmic mathematics. In particular, Algorithms and Computation in Mathematics emphasizes the computational aspects of algebraic geometry, number theory, combinatorics, commutative, non-commutative and differential algebra, geometric and algebraic topology, group theory, optimization, dynamicalsystemsandLietheory.Proposalsormanuscriptsthatcenteroncontent in non-computational aspects of one of these fields will also be regarded if the presentation gives consideration to the contents’ usefulness in algorithmic processes. More information about this series at https://link.springer.com/bookseries/3339 Oleg N. Karpenkov Geometry of Continued Fractions Second Edition 123 OlegN.Karpenkov Department ofMathematical Sciences TheUniversity of Liverpool Liverpool, UK ISSN 1431-1550 Algorithms andComputation in Mathematics ISBN978-3-662-65276-3 ISBN978-3-662-65277-0 (eBook) https://doi.org/10.1007/978-3-662-65277-0 1stedition:©Springer-VerlagBerlinHeidelberg2013 2ndedition:©Springer-VerlagGmbHGermany,partofSpringerNature2022 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. This Springer imprint is published by the registered company Springer-Verlag GmbH, DE part of SpringerNature. Theregisteredcompanyaddressis:HeidelbergerPlatz3,14197Berlin,Germany Preface to the Second Edition Theideaofthesecondeditionwasoriginallymotivatedbyimprovementofcertain notationwithinthechaptersandcorrectingvarioustypossuggestedbythereaders. However during this work I decided to add several interesting theorems that were missinginthefirstedition.Fortheconvenienceofthereaderswhoarefamiliarwith thefirsteditionIwouldliketounderlineherethemainchangesthatweremade.My intention was not to overload the book with new topics but rather to improve the expositionoftheexistingones. • FirstofallinSection1.3werelatepartialnumeratorsandpartialdenominators to the classical notion of continuants. We supplement numerous formulae via expressionsintermsofcontinuantsfurtherinthetext. • WehaveaddedacriterionofrationalanglescongruenceinSubsection2.1.8and ofintegertriangleconvergence(Proposition6.7). • In the new Section 2.5 and Section 18.6 we show the classification of integer- regularpolygonsandpolyhedrarespectively. • WehaveincludedanexplicitexpressionforLLSsequencesofadjacentanglesin termsofcertainlongcontinuedfractions(seeSection5.5). • Two algorithms to compute LLS sequences are added to Chapter 4 (see Sec- tion4.5). Finally,thechapteronGaussReductionTheory(Chapter7ofthefirsteditionof thebook)wasasubjectofthemajormetamorphose.Itwassubstantiallyrevisedand splitintoseveralnewchapters: • MarkovnumbersarediscussedinaseparateChapter7now. • Thesectionongeometryofcontinuedfractionsissubstantiallyextendedtonew Chapter8.InparticularwehaveaddedanewtechniqueofcomputationofLLS sequenceperiodsforGL(2,Z)matrices. • Chapter 9 on continuant representations of GL(2,Z) matrices is new. It is very muchinthespiritofGaussReductionTheory. v vi PrefacetotheSecondEdition • ThesemigroupofreducedmatricesisdiscussedseparatelyinChapter10. • The remaining material (of Chapter 7 of the first edition of the book) is now placedinChapter11:herewehaveaddedproofsforellipticandparabolicmatri- cesandrevisedthemaincaseofthehyperbolicmatrices.Additionallywehave extendedtheexpositiontothegroupGL(2,Z)(originallyitwasmostlyregarding SL(2,Z)). Furtherexamplesandexerciseswereaddedtodifferentchaptersofthebook. UniversityofLiverpool OlegKarpenkov February2022 Acknowledgements First, I would like to thank Vladimir Arnold, who introduced the subject of continued fractions to me and who provided me with all necessary remarks and dis- cussionsformanyyears.Second,Iamgratefultomanypeoplewhohelpedmewith remarksandcorrectionsrelatedtoparticularsubjectsdiscussedinthisbook.Among themareF.Aicardi,T.Garrity,V.Goryunov,I.Pak,E.I.Pavlovskaya,C.M.Series, M. Skopenkov, A.B. Sossinski, A.V. Ustinov, A.M. Vershik, and J. Wallner. Es- pecially I would like to express my gratitude to Thomas Garrity for exhaustively readingthroughthemanuscriptandgivingsomesuggestionstoimprovethebook. My special thanks for the amazing support from Martin Peters in particular and fromthePublisherTeamingeneral.Finally,Iamgratefultomywife,Tanya,who encouraged and inspired me during the years of working on both editions of this book. ThemajorpartofthisbookwaswrittenattheTechnischeUniversita¨tGraz.The workwascompletedattheUniversityofLiverpool.IamgratefultotheTechnische Universita¨tGrazforhospitalityandexcellentworkingconditions.Workonthisbook wassupportedbytheAustrianScienceFund(FWF),grantM1273-N18. FinallyIwouldliketothankG.R.Gerardo,R.Janssen,S.Kristensen,G.Panti, M. Peters, M. H. Tilijese, J. Wattis, M. van-Son for corrections, comments, and remarksthatwereimplementedinthesecondedition. vii Preface to the First Edition Continuedfractionsappearinmanydifferentbranchesofmathematics:thethe- ory of Diophantine approximations, algebraic number theory, coding theory, toric geometry,dynamicalsystems,ergodictheory,topology,etc.Oneofthemetamathe- maticalexplanationsofthisphenomenonisbasedonaninterestingstructureofthe setofrealnumbersendowedwithtwooperations:additiona+bandinversion1/b. This structure appeared for the first time in the Euclidean algorithm, which was known several thousand years ago. Similarly to the structures of fields and rings (withoperationsofadditiona+bandmultiplicationa∗b),structureswithaddition andinversioncanbefoundinmanybranchesofmathematics.Thatisthereasonwhy continuedfractionscanbeencounteredfarawayfromnumbertheory.Inparticular, continued fractions have a geometric interpretation in terms of integer geometry, whichweplaceasacornerstoneforthisbook. Themaingoalofthefirstpartofthebookistoexploregeometricideasbehind regularcontinuedfractions.Ontheonehand,wepresentgeometricalinterpretation ofclassicaltheorems,suchastheGauss—Kuzmintheoremonthedistributionofel- ementsofcontinuedfractions,Lagrange’stheoremontheperiodicityofcontinued fractions, and the algorithm of Gaussian reduction. On the other hand, we present somerecentresultsrelatedtotoricgeometryandthefirststepsofintegertrigonom- etry of lattices. The first part is rather elementary and will be interesting for both studentsinmathematicsand researchers.Thispartisaresultofaseriesoflecture coursesattheGrazUniversityofTechnology(Austria).Thematerialisappropriate formaster’sanddoctoralstudentswhoalreadyhavebasicknowledgeoflinearalge- bra,algebraicnumbertheory,andmeasuretheory.Severalchaptersdemandcertain experience in differential and algebraic geometry. Nevertheless, I believe that it is possibleforstrongbachelor’sstudentsaswelltounderstandthismaterial. In the second part of the book we study an integer geometric generalization of continued fractions to the multidimensional case. Such a generalization was first consideredbyF.Kleinin1895.Later,thissubjectwasalmostcompletelyabandoned due to the computational complexity of the structure involved in the calculation of the generalized continued fractions. The interest in Klein’s generalization was ix x PrefacetotheFirstEdition revived by V.I. Arnold approximately one hundred years after its invention, when computersbecamestrongenoughtoovercomethecomputationalcomplexity.After abriefintroductiontomultidimensionalintegergeometry,westudyessentiallynew questionsforthemultidimensionalcasesandquestionsarisingasextensionsofthe classical ones (such as Lagrange’s theorem and Gauss—Kuzmin statistics). This partisanexpositionofrecentresultsinthisarea.Weemphasizethatthemajorityof examplesandevencertainstatementsofthispartareontwo-dimensionalcontinued fractions. The situation in higher dimensions is more technical and less studied, andinmanycasesweformulatethecorrespondingproblemsandconjectures.The second part is intended mostly for researchers in the fields of algebraic number theory,Diophantineequationsandapproximations,andalgebraicgeometry.Several chaptersofthispartcanbeaddedtoacourseformaster’sordoctoralstudents. Finally, I should mention many other interesting generalizations of continued fractions, coming from algorithmic, dynamical, and approximation properties of continuedfractions.Thesegeneralizationsarealldistinctinhigherdimensions.We brieflydescribethemostfamousoftheminChapter27. UniversityofLiverpool OlegKarpenkov February2013