Table Of ContentAlgorithms 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
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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
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dynamicalsystemsandLietheory.Proposalsormanuscriptsthatcenteroncontent
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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
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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 con-
tinued 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
