The distribution of spacings between quadratic residues(23)

6.1.Thecasecdisc(G)>s.Weusea(h,c) c1/2+ (5.1),and (h,c) c toseethatthistermisboundedby

s 1 |λ(G)|#{sC∩L(G)}c1/2+ (6.1)rω(q)2cqc|qsupp(G)|cdisc(G)>s

BytheLipschitzprinciple(Lemma16),

#{sC∩L(G)} vol(sC)+sr 2

disc(G)

andsincevol(sC)=sr 1vol(C),we ndthat

(6.2)sr 1

#{sC∩L(G)} +sr 2

disc(G)

Moreover,inorderthatsC∩L(G)= ,wewillseethatweneedsupp(G) sr(r 1)/2,sinceCdoesnotintersectthewalls.Thisisaconsequenceofthefollowingobservation:LetC Rr 1beaboundedconvexset.De ne

r 1 diam1(C)=max{|xk|:x∈C}

k=1

Notethatdiam1scaleslinearly:diam1(sC)=sdiam1(C)foralls>0.Lemma7.Ifsupp(G)>diam1(sC)r(r 1)/2thensC∩L(G)iscontainedinthewalls{h∈Rr 1:σij(h)=0forsomei<j}.

Proof.Letdij(G)betheproductoftheprimespsuchthatσijvanishesonH(p),i.e.sothat

σij(x)=0modpforallx∈L(G)

Thendij(G)|supp(G)andmoreoverweclaimthat:

disc(G)|dij(G)

i<j

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