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ProeedingsoftheSteklovInstituteofMathematis,Vol.218,1997,pp.23{52.TranslatedfromTrudyMatematiheskogoInstitutaimeniV.A.Steklova,Vol.218,1997,pp.28{57.OriginalRussianTextCopyright 1997byArkhipov,Buriev,Chubarikov.EnglishTranslationCopyright byMAIKNauka/InterperiodiaPublishing(Russia).OnthePowerofaSingularSetinBinaryAdditiveProblemswithPrimeNumbersG.I.Arkhipov,K.Buriev,andV.N.ChubarikovReeivedFebruary,1997Binaryadditiveproblemswithprimesareproblemsonnetedwiththerepresentationofthepositiveintegersnasthesumn=a(p)+b(q).Herepandqareprimenumbers,anda(x)andb(x)arede nedinteger-valuedfuntionsofanaturalargument.Moreveritisusuallyassumedthatthenumbernissubjetedtoertainnaturaladditionalonditionsofanarithmetialnature.Theproblemofthein nityoftwinprimesinanaturalseriesandthebinaryGoldbahproblemontherepresentationofanyevennumberasthesumoftwoprimenumbers,whiharenotyetsolved,arelassialexamplesofproblemsofthiskind.Oneoftheimportantaspetsintheinvestigationofabinaryadditiveproblemistheestimationfromaboveofthedensityofitssingular(exeptional)set,i.e.,thesetofnumberswhihdonotadmitthede nedrepresentation.ThesolutionofGoldbah’sternaryproblem,whihwasfoundin1937byVinogradov[1,2℄,allowedsomeauthorstoobtain,atthesametime,the rstresultsonerningthepowerofasinglarsetforGoldbah’sbinaryproblem.Lateronmanypapersweredevotedtotheimprovementoftheseestemates.Inpartiular,in1975MontgomeryandVaughan[4℄obtained,forthe rsttime,aloweringofthepowerintheseestimatesalthoughthenumerialvalueofthisloweringwasnotomputed.AtpresentthebestresultinthisdiretionbelongstoChengJing-runandLiuJianMin[5℄.Theyprovedin1988thatfortheamoutT(x)ofevenpositiveintegersm,whihdonotexeedxandarenotrepresentableasthesumoftwoprimenumbers,asx!+1,theestimateT(x) x1 1=20wasvalid.InthisartileweestimatethepowersT1(x)andT2(x)ofsingularsetsfortwobinaryadditiveproblemswithprimes[6{8℄.Theresultsthatweobtainedanbeformulatedasthefollowingtwotheorems.Theorem1.Let 0beanirrationalnumberandletT1(x)betheamountofnumbersn xwhihannotberepresentedasn=p+[ q℄;wherepandqareprimenumbers.Then,asx!1,thefollowingestimatesarevalid:(a)ifthepartialdenominatorsofontinuedfrationsofthenumbers areboundedintheirtotality,thenT1(x) x1 2=9(lnx)8;(b)if isanalgebrainumber,thentheestimateT1(x) x1 2=9+;2324ARKHIPOVetal.where0isarbitrarilysmall,holdstrue.Theonstantunderthesign isine etive.Theorem2.Let1beanoninteger, =1.WedenotebyT2(x)theamoutofpositiveintegersn x,whihannotberepresentedasn=p+[q℄;wherepandqareprimenumbers.Then,forthequantityT2(x)wehavetheestimateT2(x) x1 2Æ+;whereisarbitrarilysmallandthequantityÆ0isde nedbytheonditionXq y e2 i [q℄ y Æ;theparameter satisfyingtheinequalityy 0:8 1 y 0:8.Inordertoprovethistheorem,weshall rstonsiderTheorem1.ProofofTheorem1.Wedenoteby thesingularsetofthegivenproblem,i.e.,theolletionofpositiveintegersywhihannotberepresentedasy=p+[ q℄:Nextwede netheintervalsP=Px,Q=Qx,Y=YxbytheonditionsPx=hx4;3x4i;Qx=hx4;x2i;Yx=h3x4;xi:Theamountofnumbersy2 ,whihsatisfytheonditiony2Yx,willbedenotedbym=m(x).Lemma1.Leta;bbenumberssubjetedtotheonditions0a1,b0.Thentheestimatem(t) t1 alnbt;whihisvalidforalltundertheonditions1t x,yieldstheestimateT1(x) x1 alnbx:Proof.Aordingtothede nition,T1(x)isthenumberofy2 ,satisfyingtheinequalityy x.Next,itisabviousthatforeveryz2[1;x℄thereexistsapositiveintegerkde nedbytheonditionz2 x 34 k;x 34 k 1 =Yx(3=4)k 1:ThereforewehavetheestimateT1(x)=m(x)+m x 34 +m x 34 2 + x1 alnbx 1+ 34 1 a+ 34 2(1 a)+ x1 alnbx:PROCEEDINGSOFTHESTEKLOVINSTITUTEOFMATHEMATICSVol.2181997ONTHEPOWEROFASINGULARSETINBINARYADDITIVEPROBLEMS25Wehaveprovedthelemma.ItfollowsfromLemma1thatinordertoproveTheorem1,itsuÆestoobtainorthequantitym(x)essentiallythesameestimatewhihisrequiredforT1(x).Inthisonnetion,weonsidertheDiophantineequationy=p+[ q℄undertheonditionthat,asbefore,y2 andpandqareprimenumbers.Inthisasewherey2 ,thenumberofsolutionsofthisequationiszero.Ontheotherhand,weanformallyexpressthisquantityintermsoftrigonometrisums.Weassumethaty2Yx,p2Px,[ q℄2Qx.Then,obviously,wehavetherelationJ=1Z0SVWd ;whereS=S( )=Xp Pxe2 i plnp;V=V( )=X[ q℄2Pxe2 i [ q℄lnq;W=W( )=Xy2 ;y2Yxe2 i y:Nextweset =x7=9,{= 1,E1=[ {;{℄,E2=[{;1℄.Then,byvirtueoftheperiodiityofthefuntionSVWwehave0=1Z0SVWd ={Z { +1Z{ =J1+J2;wherethequantitiesJ1andJ2areobviouslyde nedbythelastrelation.ItislearthatjJ1j=jJ2j.TheplanofourfurtherresoningonsistsinanappropriateestimationfrombelowthequantityjJ1joftheformjJ1j mx;andtheestimationfromaboveofthequantutyjJ2joftheformjJ2j m1=2x1=2 ;wherem=m(x), =x8=9(lnx)4.Itiseasytoseethat,indeed,theseestimatesyieldtherequiredestimateforthequantitym.Weshall rstonsidertheestimatejJ2j.WedenotebyF( )thequantityF( )=min jS( )j;jV( )j :Then,obviously,forall wehavejS( )V( )j F( ) jS( )j+jV( )j :ItfollowsthatjJ2j= 1 {Z{SVWd max 2E2F( ) 1Z0jS( )W( )jd +1Z0jV( )W( )jd :PROCEEDINGSOFTHESTEKLOVINSTITUTEOFMATHEMATICSVol.218199726ARKHIPOVetal.Next,usingtheCauhyinequality,weget1Z0jSWjd 1Z0jSj2d 1Z0jWj2d 1=2=Xp2Pxln2p Xy2 y2Yx1!1=2 (mxlnx)1=2:Usingsimilararguments,wederiveanestimateoftheform1Z0jVWjd (m