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LatticeRulesbyComponentScaling toconstructfamiliesoflattices.WedeterminetherelationbetweentheZaremba Weintroduceatheoryofrectangularscalingofintegerlattices.Thismaybeused J.N.Lynessy AbstractT.Srevikz AMSSubjectClassication:65D3206B99 index()ofvariousmemb
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@MISC{_latticerulesbycomponentscalingtoconstructfamiliesoflattices.wedeterminetherelationbetweenthezaremba,
author = {},
title = {LatticeRulesbyComponentScaling toconstructfamiliesoflattices.WedeterminetherelationbetweentheZaremba Weintroduceatheoryofrectangularscalingofintegerlattices.Thismaybeused J.N.Lynessy AbstractT.Srevikz AMSSubjectClassication:65D3206B99 index()ofvariousmemb},
year = {}
}
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Abstract
Intermsofthiswehavethefollowingdenition. where ThisworkwassupportedbytheAppliedMathematicalSciencessubprogramoftheOceofEnergy xi=max(jxij;1): (x)=x1x2:::xs; (1:2) (1:1) Research,U.S.DepartmentofEnergy,underContractW-31-109-Eng-38,andbytheNorwegianCouncil forHumanitiesandScience. zDept.ofInformatics,UniversityofBergen,Norway yMathematicsandComputerScienceDivision,ArgonneNationalLaboratory,Argonne,IL 1latticeis Denition2TheZarembaindexorgureofmerit,(),ofans-dimensionalinteger quadraturerulebasedonQ,thelatticereciprocalto.ThislatticeruleemploysallN sets.TheZarembaindex,(),isarecognizedmeasureofeciencyofthemultidimensional Notethatalllatticepointsofanintegerlatticehaveintegercomponents.Thus(x)and ()arepositiveintegers.InSection3weshallgeneralizethisdenitiontootherpoint ()=min x2;x6=0(x): (1:3) pointsofQlyingin[0;1)s,Nbeingtheorderof.Thus,attentionhasbeendevoted tosearchinginLs(N),thesetofs-dimensionalintegerlatticesoforderN,withaviewto cancomparelatticeshavingdierentvaluesofN.OurmeasureisbasedontheZaremba conjecturethatsuggeststhatthereexistsaconstantzssuchthat ndingoptimallatticesofthisorder,thatis,lattices0forwhich(0)coincideswith Itisconvenienttointroducea\measureofgoodness"ofalatticeusingwhichone s(N)zsN s(N)=max (logN)s�2;s2: 2Ls(N)(): (1:4) Fors=2theconjecturehasbeenprovedbyZaremba[Zar66].FollowingKedemand Zaremba[KZ74],weassigntoeverylatticeavaluedenedby z()=()(logN)s�2 N: (1:5)
Keyphrases
j.n.l ynessy abstractt srevikz amssubjectclassication min x2 norway ymathematicsandcomputersciencedivision ofans-dimensionalinteger quadraturerulebasedonq coincideswith itisconvenienttointroducea measureofgoodness ofalatticeusingwhichone thislatticeruleemploysalln set u.s.d epartmentofenergy followingkedemand zaremba kz74 xi max andbythenorwegiancouncil forhumanitiesandscience withaviewto cancomparelatticeshavingdierentvaluesofn
