The distribution of spacings between quadratic residues(26)

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

THEDISTRIBUTIONOFSPACINGSBETWEENQUADRATICRESIDUES21

Nowthesumc|dc1/2+ isboundedbyτ(d)d1/2+ d1/2+ ,sothe

aboveisboundedby d 1/2+ s 1/2+

d|q

d>s

byLemma19.Thisboundsthecontributionofc,Gwithcsupp(G)>s.

sIfcdisc(G)>sthen≤c.This,togetherwith(6.6)impliesthat1 1/2+ ssupp(G) csdisc(G)qc|qsupp(G)|cdisc(G)>s

csupp(G)<s

1 1/2+ 1/2+ cg s1 sqc|qg|cg<sc|qg|qc<sg<s

s

since 1/2+ 1 s 1/2+ , c|qc<s 2

(Notethatweonlyusedsupp(G) sr(r 1)/2tobound(6.5)!)

6.2.Thecasecdisc(G)≤s.Fixc≥1andGandpartitionthelatticepointsinsC∩L(G)intotwosubsetsasfollows:Fixareducedfunda-mentalcell(seeB.1)P=P(G)forthelatticeL=L(G).ThencPisareducedfundamentalcellforthedilatedlatticecL.WecantileRr 1bythetranslateshc+cP,hc∈cL.

De nition6.1.Wesaythatx∈L∩sCisc-interiorifthereissomey∈cLsothatx∈y+cP sC.Wesaythatx∈L∩sCisac-boundarypointotherwise.

NotethatthenotiondependsoncandonthechoiceofafundamentalcellPforL.

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