Diophantine Approximation: Festschrift for Wolfgang Schmidt

415 Pages·2008·2.45 MB·English

by  Robert F. Tichy

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
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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