The distribution of spacings between quadratic residues

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

THEDISTRIBUTIONOFSPACINGSBETWEEN

QUADRATICRESIDUES

¨KURLBERGANDZEEV´RUDNICKPAR

Abstract.Westudythedistributionofspacingsbetweensquaresmoduloq,whereqissquare-freeandhighlycomposite,inthelimit

asthenumberofprimefactorsofqgoestoin nity.Weshowthat

allcorrelationfunctionsarePoissonian,whichamongotherthings,

impliesthatthespacingsbetweennearestneighbors,normalizedto

haveunitmean,haveanexponentialdistribution.

Date:Dec14,1998.

SupportedinpartbyagrantfromtheIsraelScienceFoundation.Inaddition,the rstauthorwaspartiallysupportedbytheECTMRnetwork”AlgebraicLieRepresentations”,EC-contractnoERBFMRX-CT97-0100.

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