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1、APROOFOFTHEMATRIXVERSIONOFBAKER’SCONJECTUREINDIOPHANTINEAPPROXIMATIONTUSHARDASANDDAVIDSIMMONSABSTRACT.WeprovethatthematrixanalogueoftheVeronesecurveisstronglyextremalinthesenseofDiophantineapproximation,therebyresolvingaquestionposedbyBeres-nevich,Kleinbock,andMargulis(’15)intheaffirmative
2、.1.INTRODUCTIONTheoriginsofmetricDiophantineapproximationonmanifolds(alsoknownasDio-phantineapproximationwithdependentquantities)maybetracedbacktoKurtMahler’sprofoundconjectureintranscendencetheory,[12],thatmaybetranslatedthus:LebesguealmosteverypointontheVeronesecurve{(x,x2,...,xn):x∈R}i
3、snotverywellap-proximablebyrationalvectors.Mahler’sconjecturewasresolvedthreedecadeslaterbyVladimirSprindˇzuk[13],whowentontoconjecture[14]thatanyanalyticnondegener-atesubmanifoldofRnisstronglyextremal,viz.thatLebesguealmosteverypointisnotverywellmultiplicativelyapproximablebyrationalvect
4、ors.AspecialcaseofSprindˇzuk’sconjecturewasconjecturedearlierbyAlanBaker,[2,p.96],viz.thattheVeronesecurveisstronglyextremal.Wereferthereaderto[5]formoreregardingthishistoryandallieddevelopments.Afteraslewofpartialresultsbyanumberofauthors,Sprindˇzuk’sconjecturewasfi-nallyresolvedbyDmitryK
5、leinbockandGrigoryMargulisintheirlandmark1998AnnalsarXiv:1510.09195v3[math.NT]22Mar2018paper[9],wheretheytranslatedtheproblemintodynamicaltermsandsuccessfullylever-agedquantitativenon-divergenceestimatesforunipotentflowsonhomogenousspaces.TherehassincebeenanintensestudyofDiophantineextrema
6、lity,bothinthecaseofsimultaneousapproximation[8,15,16]andinthematrixapproximationframework[7,11,4,1,6].Inparticular,inajointworkwithJunboWang,KleinbockandMargulisgeneralizedtheirresulttothematrixcasebyprovingthatanystronglynonplanarman-ifoldinthespaceofmatricesisstronglyextremal[11,Theore
7、m2.1](cf.Definition1.2below).Strongnonplanarityisnotonlyastrongerrequirementthannondegeneracy,but2010MathematicsSubjectClassification.11J83,11J13,11K60.Keywordsandphrases.SimultaneousDiophantineapproximation,linearforms,metrictheory,Diophantineapproximationonmanifolds
