DIOPHANTINE APPROXIMATION Festschrift for Wolfgang Schmidt Developments in Mathematics VOLUME 16 Series Editor: Krishnaswami Alladi, University of Florida, U.S.A. Aims and Scope Developments in Mathematics is a book series publishing (i) Proceedings of conferences dealing with the latest research advances, (ii) Research monographs, and (iii) Contributed volumes focusing on certain areas of special interest. Editors of conference proceedings are urged to include a few survey papers for wider appeal. Research monographs, which could be used as texts or references for graduate level courses, would also be suitable for the series. Contributed volumes are those where various authors either write papers or chapters in an organized volume devoted to a topic of special/current interest or importance. A contributed volume could deal with a classical topic that is once again in the limelight owing to new developments. DIOPHANTINE APPROXIMATION Festschrift for Wolfgang Schmidt Edited by HANS PETER SCHLICKEWEI Philipps-Universität Marburg, Marburg, Germany KLAUS SCHMIDT Universität Wien, Vienna, Austria ROBERT F. TICHY Technische Universität Graz, Graz, Austria 2000 Mathematics Subject Classification: 11D, 11J, 11K This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machines or similar means, and storage in data banks. Product Liability: The publisher can give no guarantee for the information contained in this book. This also refers to that on drug dosage and application thereof. In each individual case the respective user must check the accuracy of the information given by consulting other pharmaceutical literature. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. © 2008 Springer-Verlag/Wien Printed in Germany SpringerWienNewYork is part of Springer Science+Business Media springer.at Typesetting: Scientific Publishing Services (P) Ltd., Chennai, India Printing: Strauss GmbH, Mörlenbach, Germany Printed on acid-free and chlorine-free bleached paper SPIN 12102310 With 10 figures Library of Congress Control Number 2008930106 ISBN 978-3-211-74279-2 SpringerWienNewYork e-ISBN 978-3-211-74280-8 SpringerWienNewYork CONTENTS Preface vii The mathematical work of Wolfgang Schmidt 1 Hans Peter Schlickewei Schäffer’s determinant argument 21 Roger C. Baker Arithmetic progressions and Tic-Tac-Toe games 41 József Beck Metric discrepancy results for sequences {n x} and Diophantine equations 95 k István Berkes, Walter Philipp, and Robert F. Tichy Mahler’s classification of numbers compared with Koksma’s, II 107 Yann Bugeaud Rational approximations to a q-analogue of πand some other q-series 123 Peter Bundschuh and Wadim Zudilin Orthogonality and digit shifts in the classical mean squares problem in irregularities of point distribution 141 William W. L. Chen and Maxim M. Skriganov Applications of the Subspace Theorem to certain Diophantine Problems: Asurvey of some recent results 161 Pietro Corvaja and Umberto Zannier Ageneralization of the Subspace Theorem with polynomials of higher degree 175 Jan-Hendrik Evertse and Roberto G. Ferretti On the Diophantine equation G (x) =G (y) with Q(x, y) =0 199 n m Clemens Fuchs, Attila Petho´´,and Robert F. Tichy Acriterion for polynomials to divide infinitely many k-nomials 211 Lajos Hajdu and Robert Tijdeman Approximants de Padé des q-polylogarithmes 221 Christian Krattenthaler et Tanguy Rivoal The set of solutions of some equation for linear recurrence sequences 231 Viktor Losert Counting algebraic numbers with large height I 237 David Masser and Jeffrey D. Vaaler vi Contents Class number conditions for the diagonal case of the equation of Nagell and Ljunggren 245 Preda Miha˘lescu Construction of approximations to zeta-values 275 Yuri V. Nesterenko Quelques aspects diophantiens des variétés toriques projectives 295 Patrice Philippon et Martín Sombra Une inégalité de Lojasiewicz arithmétique 339 Gaël Rémond On the continued fraction expansion of a class of numbers 347 Damien Roy The number of solutions of a linear homogeneous congruence 363 Andrzej Schinzel Anote on Lyapunov theory for Brun algorithm 371 Fritz Schweiger Orbit sums and modular vector invariants 381 Serguei A. Stepanov New irrationality results for dilogarithms of rational numbers 413 Carlo Viola PREFACE This volume contains 22 research and survey papers on recent developments in the field of diophantine approximation. The first article by Hans Peter Schlickewei is devoted to the scientific work of Wolfgang Schmidt. Further contributions deal with the subspace theorem and its applications to diophantine equations and to the study of linear recurring sequences. The articles are either in the spirit of more classical diophantine analysis or of geometric or combinatorial flavor. In particular, estimates for the number of solutions of diophantine equations as well as results concerning congruences and polynomials are established. Furthermore, the volume contains transcendence results for special functions and contributions to metric diophantine approximation and to discrepancy theory. The articles are based on lectures given at a conference at the Erwin Schrödinger Institute in Vienna in 2003, in which many leading experts in the field of diophantine approximation participated. The editors are very grateful to the Erwin Schrödinger Institute and to the FWF (Austrian Science Fund) for the financial support and they express their particular thanks to Springer-Verlag for the excellent cooperation. Robert F. Tichy DiophantineApproximation H.P.Schlickeweietal.,Editors ©Springer-Verlag2008 THE MATHEMATICAL WORK OF WOLFGANG SCHMIDT HansPeterSchlickewei FachbereichMathematikundInformatik,Philipps-UniversitätMarburg,Hans-Meerwein-Strasse,35032 Marburg,Germany [email protected] Introduction Wolfgang Schmidt’s mathematical activities started more than fifty years ago in 1955.Inthemeantimehehaswrittenmorethan180papers–manyofthemcontaining spectacularresultsandbreakthroughsindifferentareasofnumbertheory. StudyingthelistofhispublicationswemayclassifyWolfgangSchmidt’sscientific papersunderthefollowingheadings:(1)geometryofnumbers,(2)uniformdistribu- tion,(3)approximationofrealnumbers,(4)heights,(5)approximationofalgebraic numbers(qualitativeresults),(6)normformequations(qualitativeresults),(7)tran- scendentalnumbers,(8)elementaryproofoftheRiemannhypothesisforcurves,(9) nonlinearapproximationofrealnumbers,(10)zerosandsmallvaluesofforms,(11) quadratic geometry of numbers, (12) approximation of algebraic numbers – quanti- tativeresults,(13)normformequations–quantitativeresults,(14)linearrecurrence sequences. TheorderingofthislistischronologicalaccordingtothedatewhenSchmidthas writtenhisfirstpaperontherespectivesubject.Inthesequelwewilldiscussforeach of these subjects one of the important results obtained by Schmidt in the respective area.InviewofthelargenumberofoutstandingpaperswrittenbyWolfgangSchmidt, the choice was rather difficult. It certainly depends also upon personal taste and no doubtformostsubjectsalsoadifferentchoicewouldhavebeenwelljustified. 1 Geometryofnumbers (Schmidt’spapers[1–3,5–7,9,10,12,14,21,24,32,40,65,91]dealwiththissubject.) Schmidt’sfirstmathematicalpaper[1]appearedin1955,whenhewasnotyet22 years old. It is his doctoral dissertation, which was written under the guidance of his supervisor Edmund Hlawka inVienna. Recall Minkowski’s famous latticepoint theorem: LetS ⊂RnbeasymmetricconvexbodyofvolumeV(S).Letmbeanaturalnumber. Thenforanylattice(cid:1)inRn withdeterminantd((cid:1))=d satisfying 1 d < V(S) (1.1) m2n Scontainsm distinctpairsofpoints±u ,...,±u ∈(cid:1)\{0}. 1 m 2000Mathematicssubjectclassification. 11-02. 2 H.P.Schlickewei IntheoppositedirectionHlawka(1943)hadshownthefollowing: LetS ⊂Rn beaboundedsymmetricstarbodyofvolumeV(S). Thenforanyd satisfying 1 d < ζ(n)−1V(S) (1.2) 2 thereexistsalattice(cid:1)inRn ofdeterminantd((cid:1))=d suchthat Scontainsnopoint u∈(cid:1)\{0}. Inhisthesis[1]SchmidtwasabletoimproveuponHlawka’sresult.Moreover,he gotthefollowingextension: Foranynaturalnumberm ≥m andforanyd satisfying 0 1 d < ζ(n)−1V(S) (1.3) 2m there exists a lattice (cid:1) of determinant d((cid:1)) = d such that S does not contain m distinctpairsofpoints±u ,...,±u ∈(cid:1)\{0}. 1 m WementionthatCasselsinhismonograph[C2]dedicatedseveralpagestoSchmidt’s thesis. 2 Uniformdistribution InviewofhisoriginsfromEdmundHlawka’sViennaschoolofmathematicsitseems to be quite natural that a topic of recurrent interest in Schmidt’s scientific activities have been problems from the theory of uniform distribution. His series of papers [39,43–45,48,63,66,73,74,82] as well as his lecture notes [86] certainly should be mentionedinthiscontext. HerewequotetwospectaculartheoremsonuniformdistributionprovedbySchmidt: Foraninfinitesequenceofpointsx ,x ,...inthek-dimensionalunit-cube[0,1)k, 1 2 for N ∈ N and for a point α = (α1,...,αk) with 0 < ακ ≤ 1 (κ = 1,...,k) we write A(α,N) for the number of points x with 1 ≤ n ≤ N satisfying x ∈ n n [0,α1)×...×[0,αk).Thediscrepancy DN ofthesequence{xn}n∈Nthenisdefined as (cid:1) (cid:1) DN = sup (cid:1)(cid:1)(cid:1)A(α,N) −α1·...·αk(cid:1)(cid:1)(cid:1). (2.1) α∈(0,1]k N K.F.Roth[R1]provedin1954: Supposek ≥1.Thereexistsapositiveconstantc dependingonlyuponk withthe k followingproperty: Foranyinfinitesequence{xn}n∈N asabovethereareinfinitelymanynaturalnum- bers N suchthatwehave D ≥c N−1(logN)k/2. (2.2) N k In1972Schmidt[66]succeededtoobtaininthecasek =1thefollowingimprove- mentof(2.2): ThemathematicalworkofWolfgangSchmidt 3 Thereexistsapositiveconstantcwiththefollowingproperty: For any infinite sequence {xn}n∈N in the unit interval [0,1) there exist infinitely manynaturalnumbers N suchthatwehave D >cN−1logN. (2.3) N Asforaninequalityintheoppositedirection,itiswellknownthatforanyirrational realnumberαwhichhasboundedpartialquotientsinitscontinuedfractionexpansion thesequence{nα}(where{x}denotesthefractionalpartofx)satisfies logN D ≤c(α) . N N Herec(α)isapositiveconstantdependingonlyuponα.Therefore(2.3)isbestpossible. InalaterpaperSchmidtshowsinamostspectacularwaythatresultsoftype(2.2)or (2.3)dependverymuchuponthesetsusedinthedefinitionofdiscrepancy.In(2.1)we usesubcubesoftheunitcubewhoseaxesareparalleltothecoordinateaxestodefine the discrepancy. If we allow rotations, the result changes dramatically. We quote in thiscontextSchmidt’sresultfrom1969[45]: Fork ∈Nwelet Sk ={x ∈Rk+1|x2+...+x2 =1} 0 k bethek-dimensionaltheunit-sphere. OnSk weintroducethenormalizedLebesgue-measureσ withσ(Sk) = 1.Fora sequenceofpoints x ,...,x inSk andforasubset M ofSk wewrite A(M)for 1 N thenumberofpointsx ∈ M (i =1,...,N).NowSchmidt’sresultreadsasfollows: i Letx ,...,x beany N pointsinSk.Thenforanyε >0thereexistsaspherical 1 N cap K inSk suchthat (cid:1) (cid:1) (cid:1)(cid:1)(cid:1)A(K) −σ(K)(cid:1)(cid:1)(cid:1)>c(n,ε)N−1/2−1/2k−ε. (2.4) N Beck[Be]hasimprovedthebound(2.4).Indeedhegotridofthe−εintheexponent. Moreover,heshowedthatapartfromtheε,theexponentin(2.4)isbestpossible. 3 Approximationofrealnumbers Dirichlet’sTheoremonrationalapproximationofrealnumberssaysthefollowing: Suppose α is an irrational real number. Then there are infinitely many rational numbers p/q suchthatwehave (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:1)α− p(cid:1)(cid:1)<q−2. (3.1) q Foranalgebraicnumberβ ofdegreed let P(X)bethedefiningpolynomialofβ overQ,i.e.,theuniqueirreduciblepolynomialwithcoprimeintegralcoefficientsand positiveleadingcoefficienthaving P(β)=0.Write H(β)forthemaximumabsolute valueofthecoefficientsof P.ThefollowingconjecturewouldgeneralizeDirichlet’s