The distribution of spacings between quadratic residues(6)

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

THEDISTRIBUTIONOFSPACINGSBETWEENQUADRATICRESIDUES5writereff(h)asalinearcombinationofcharacteristicfunctionsofcer-tainhyper-planesoverZ/pZ.Next,insection5weusethemultiplica-tivepropertiesofthecountingfunctionsN(h,q)toderiveanexpressionforRr(C,q)asasumoverdivisorscofqandlatticesLarisingfromintersectionsofhyper-planesmodulopfordi erentp’s(proposition6).Insection6weshowthatthemaintermofthesumconsistsofthosetermsforwhichtheproductofcandthediscriminantofLaresmallwithrespecttos,andanerrortermcorrespondingtotermswheretheproductislarge.Insection7weevaluatethemaintermandshowthatitgivesusexactlyvol(C),thusgivingusourmainresult.

InappendixAweexplainhowtouseTheorem1toderiveresultssuchasTheorem2,thatthelevelspacingsarePoissonianaswell.Appen-dixBexplainssomebackgroundoncountinglatticepointsinconvexsets.InappendixCweestimatethenumberofdivisorsofqthataresmallerthana xedpowerofthemeanspacings.

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