The distribution of spacings between quadratic residues(5)

Abstract. We study the distribution of spacings between squares modulo q, where q is square-free and highly composite, in the limit as the number of prime factors of q goes to infinity. We show that all correlation functions are Poissonian, which among oth

4¨KURLBERGANDZEEV´RUDNICKPAR

Theorem2.Forqsquare-free,thelimitinglevelspacingdistributionofthesquaresmoduloqisgivenbyP(t)=exp( t)asω(q)→∞.Moreover,underthesamecondition,foranyk ≥1thelimitingjointdistributionof(sn,sn+1,...,sn+k)isaproductk

i=0exp( ti)ofk+1

independentexponentialvariables.

Thereareonlyafewknowncaseswherethecompletespacingdis-tributioncanbeprovedtobePoissonianasinourcase.AnotableexampleisHooley’sresults[7,8,9,10]thatthespacingsbetweenele-mentsco-primetoqarePoissonianasthemeanspacingq/φ(q)→∞.AmuchmorerecentresultisduetoCobeliandZaharescu[2]whoshowthatthespacingsbetweenprimitiverootsmoduloaprimeparePoissonianprovidedthemeanspacingp/φ(p 1)→∞.

Theresultsofthispaperarerelatedtoworkonthelevelspacingdistributionofthefractionalparts{αn2}(αirrational)byRudnick,SarnakandZaharescu[16,17].Inparticular,in[17]anattempttostudythatproblemismadebyreplacingαwitharationalapproxi-mationb/q,andthisleadstostudythespacingsofthesequencebn2modq,1≤n≤NforNasmallpowerofq.Theavailablesitesareexactlythesetofsquaresmoduloq,andhenceourinterestintheproblem.

In[17],itisshownthatinorderforallthecorrelationfunctionsofthesequence{αn2}tohavePoissonbehavior,itisnecessarytoassumethattherationalapproximantsb/qhavedenominatorqwhichisclosetosquare-free.Henceourinterestinthesquare-freecase.ForarbitraryqitisstilltruethatallcorrelationsarePoissonian,buttherearesigni canttechnicalcomplicationstoovercomeinprovingthis,see

[13].

Webelievethatthemethodsdevelopedinthispapershouldbeusefulinstudyingsimilarproblems,forinstancethespacingdistributionofcubesmoduloq,asthenumberofprimefactorsofqthatarecongruentto1modulo3tendstoin nity.(Theconditionmodulo3isnecessaryinorderforthemeanspacingtogotoin nity.)

Contentsofthepaper:Webeginwithasectionsketchingtheargu-mentforTheorem1inthecaseofthepaircorrelationfunction.Thissectioncanbeusedasaguidetotherestofthepaper.

Insection3we rstreducetheproblemtothecasethatqisodd.Theninsection4weanalyzethebehaviorofN(h,p)wherepisprime.Squaresthataredistinctmoduloqarenotnecessarilydistinctmodulop;wedenotebyreff(h)inganinclusion-exclusionargumentwe

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