Applications of Dynamical Systems in Combinatorial Number Theory - Yihan Zhang.pdf

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1、ApplicationsofDynamicalSystemsinCombinatorialNumberTheoryYihanZhangJanuary16,2016Contents1Introductiontodynamicalsystems22Dynamicalsystemsvs.numbertheory53F•urstenberg'sproofofSzemeredi'stheorem74Gowers'proofofSzemeredi'stheorem10F•urstenberg(1977)gaveanewproofofSzem

2、eredi'stheorem,bringingtopologicaldy-namicsintocombinatorialnumbertheory,promotinganewspecialization.Dynamicalsys-temshavevariousbranches,e.g.,topologicaldynamicalsystems,ergodicdynamicalsystems,smoothdynamicalsystems,randomdynamicalsystems,Hamiltoniandynamicalsystems

3、,etc.Nowadays,asfortechniquesindynamicalsystems,itistopologicaldynamicsander-godictheorythataremostusedincombinatorialnumbertheory.Inthisnote,weconsiderapplicationsofdynamicalsystemsincombinatorialnumbertheory,includingfoursections.1.Abriefintroductiontodynamicalsystem

4、sincludingbasicconceptsandresults,inwhichwewillseethatdynamicalsystemsbearastrongrelationshipwithcombinatorialnumbertheory;2.applicationsofdynamicalsystemsincombinatorialnumbertheory,F•urstenberg'snewproofofvanderWaerden'stheoremviatopologicaldynamicsandhisnewproofofSz

5、emeredi'stheoremviaergodictheory;3.abriefsketchofF•urstenberg'sproofofSzemeredi'stheorem;4.basicideasofGowers'proofofSzemeredi'stheoremviahigherorderFourieranalysis.11IntroductiontodynamicalsystemsDynamicalsystemsisasubjectconcerningwithqualitativepropertiesofgroupa

6、ctionsonspaces,wheregroupactionsaremapssubjectto1.e(x)=x;2.(g1g2)(x)=g2(g1(x)).ConsideractionsofsemigroupZ+andgroupZ.1.Z+-action.T:X!X,T0=Id;T1=T;T2=TT;


.2.Z-action.T:X!X,


;T1;T0=Id;T1=T;T2=TT;


.Denition1(t.d.s.,Birkho).(X;G)iscalledat.d.s.ifXisacompactmetri

7、cspaceandGisatopologicalgrouporsemigroupactingonX.Forexample,G=Z+;(X;T):=(X;Z+);T:X!Xisacontinuousmap.G=Z;(X;T):=(X;Z);T:X!Xisahomeomorphism.WewillalsoneedZd-actionsubjecttoTiTj=TjTi;1ijd:Denition2(m.d.s.,Poincare,Birkho,vonNeumann).(X;B;;G)iscalledanm.d.s.ifX

8、isaset,Bisa-algebraonX,Gisagrouporsemigroup,andg:X!Xisameasure-preservingtransformationsubjectto(B)=(g1(B));8B2B;