Structural Additive Theory

425 Pages·2013·4.16 MB·English

by  David J. Grynkiewicz (auth.)

Developments in Mathematics
VOLUME30
SeriesEditors:
KrishnaswamiAlladi,UniversityofFlorida,Gainesville,FL,USA
HershelM.Farkas,HebrewUniversityofJerusalem,Jerusalem,Israel
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David J. Grynkiewicz
Structural
Additive
Theory
DavidJ.Grynkiewicz
InstitutfürMathematikundWissenschaftliches
Rechnen
Karl-Franzens-UniversitätGraz
Graz,Austria
ISSN1389-2177 DevelopmentsinMathematics
ISBN978-3-319-00415-0 ISBN978-3-319-00416-7(eBook)
DOI10.1007/978-3-319-00416-7
SpringerChamHeidelbergNewYorkDordrechtLondon
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MathematicsSubjectClassification(2010): 11P70,11B75,11B50,20K01,11H06
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Preface
ThisbookgrewoutofseveralcoursesgivenatKarl-Franzens-UniversitätGrazin2008–2012
andhassincebeenaugmentedwithadditionalmaterial.Itshouldbeofinterestbothtopeople
newtothefieldvariouslyknownasAdditiveNumberTheory,AdditiveCombinatorics,Additive
GroupTheoryandCombinatorialNumberTheory—asabasicintroductiontothearea—aswell
as to the more seasoned researcher, in view of the unified presentationof material previously
onlyavailableinresearcharticlescombinedwithafairamountofnewmaterial.Asthereseems
no real consensus on whether additive problems belong to Combinatorics, Number Theory,
Group Theory, or even Analysis, as they often find themselves lying between all these more
establishedareas of mathematics,we willrefer to the broaderarea of mathematicsdealtwith
bythisbooksimplyasAdditiveTheory.
Inrecentyears,thefirstfewcomprehensivetextsonwhathasbecomearapidlydeveloping
subjecthavebeguntobepublishedinAdditiveTheory.Withsofewtextsonthesubject,Ihave
madelittleattempttocompetewiththesealreadyestablishedworks.Indeed,thefocusofthis
text is specifically on those areas of Additive Theory that have not been treated in detail by
previous books, and even when treating more basic results also found in other texts, I have
endeavoredtopresentsuchresultseitheringreatergeneralityorwithnewproofvariations.
Rather than focus on the great achievements in approximate results—such as Freiman’s
Theorem, Szemeredi’s Theorem, and results achieved via Fourier Analytic/Ergodic Theory
breakthroughs—resultsthatrequirelittlehypothesesbut,atthisprice,yieldonlyroughresults
withimpreciseconstantsthatleavemuchroomforfutureimprovement,Ihaveinsteadfocused
on the more exact and refined results in the area, results which have some satisfactory air of
completenesstothemselves,eveniftheymayyetonedayfitintoawiderlandscape.Thisre-
flectsbothapersonalbiasaswellastheaforementionedwishtocomplement,andnotcompete
with,thecurrentmaterialavailable.
Ontopofthis,Ihavenotshiedawayfrompresentingimportantresultsevenwhentheyhave
notoriouslycomplicatedproofs.Thereislittleprerequisiteforthisbookapartfromasolidback-
ground in undergraduate mathematics, particularly in the theory of finitely generated abelian
groupsandLinearAlgebra,whichmakesthematerialsuitableforagraduateoradvancedun-
dergraduatecourse.Knowledgeofthebasicalgebraicconceptsofgroups,rings,fields,vector
spacesandmodulesisassumed,thoughverylittlebeyondthedefinitionsandbasicswouldbe
needed. The main prerequisite might best be described as a fair amount of mathematical ma-
turity. I suspect that the intricacy of the more difficult proofs treated here may be the like of
which a less experienced student has not yet seen. For this reason, I have taken great care to
include as many details as reasonable, many more than is common in other texts, when pre-
sentingeachandeveryproof.Whilethismayexpandthelengthofeachproofinprint,Ihope
thatitwillalsoreducethetimeneededtoabsorbthecomplicatedcombinatorialargumentsthat
oftenarise.Manyofthechaptersareselfcontainedorcomeinaseriesofinterconnectedtopics,
v
vi Preface
whichgivesthelecturerabitoffreedomintheselectionofchapterstopresent.Unlikemany
othertexts, if a theoremis notquotedina givensection, itis not used,whichshouldmakeit
quiteeasyforalecturertoperusethematerialanddecidewhattopresent.
While many mathematical subjects are marked by a slew of tiny lemmas and propositions
slowlybuildingupsufficientmachinerytobegintacklinglargerproblems,AdditiveTheoryis
marked by having its fundamental results more like solitary columns, each of which must be
summitted by whatever creative means is available. Once enough such results are proven, it
slowlybecomespossibletobuildupthemoregeneraltheory,aprocesswhichisstillunderway,
tosolveotherproblemsin,andoutof,thefield.Forthistext,Ihaveoftenchosentopresentthe
morefoundationalresults,ratherthantheirapplications,collectingtogetherthoseresultswhich
can be used to prove other results in the field and further. Putting the theorems into practice
hasbeenleftmostly,thoughnotentirely,totheexercises.Allthisresultsinaslightlydifferent
structure, namely, many chapters are devoted to proving one or two key results rather than
presentingrelatedresultsonacommontheme.Thethematicgroupingoccursatamuchhigher
level,andthebookcanbedividedintothreemajorparts:onedealingwithsumsetsofsets,one
withsubsequencesumsofsequences,andoneintroducingsomeofthemoreadvancedmethods
from the field. For the latter, a fair amount of linear algebra and the occasional other random
subjectishelpful.However,whileIhavenotendeavoredtoproveeverybasicresultusedinthe
proofs,Ihaveattemptedtomakeitclearexactlywhichresultisbeingusedatanygivenpoint.
Thus,astudentlessfamiliarwiththesemoreadvancedtopicscansimplytaketheprerequisites
here as given, and this may actually facilitate their absorbtion when seen in a later course on
thesubject.Regardless,thisisarareoccurrence,andthefewnon-selfcontainedpre-requisites
forthisfinalpartcanbequicklyreviewedonthesideifneeded.
For the seasoned researcher, there is quite a bit more in this text than you might expect.
Indeed, many chapters contain entire new proofs, some more major than others, for classical
results. At other times, older results have been given slight generalizations, and many new
resultsnotfoundinothertextshavebeencollectedhere.Thefollowingaresomehighlights:
• AnentireextendedtheoryofFreimanhomomorphism(cid:2)s,developednotjustforn-foldsumsets
nA, but also for sumsets having distinct summands n A . Weaker, more flexible forms
i=1 i
of restricted homomorphisms are also treated. The universal ambient group is introduced
in normalized form, an upper bound on its torsion subgroup is given, several short exact
sequences are derived, and its value is explicitly calculated for small torsion-free sumsets
(belowthe3k−4bound)andforgeneralsumsetsbelowtheCauchy-Davenportbound.
• The entire theory of finite sumsets is extended to include certain infinite summands. This
novelviewpointgivesanaturalwaytoviewcontainmentby,andcontainmentof,arithmetic
progressionsastwocasesoftheverysamephenomenon.
• AnentirechapterdevotedtothebasicsoftheIsoperimetricmethod(oftenneglectedinother
texts),includingapplicationstodirectadditivequestionsinvolvingSidonsetsshowingthata
Sidonset(fromanarbitraryabeliangroup)musthavelargesumsetwithanyotherset.
• AsimplifiedproofofKemperman’sStructureTheorem—whichisoftenavoidedsolelyow-
ingtoitscomplexity—basedofftherecentideasusedtoextendit.
• TheDeVos-Goddyn-MoharTheorempresentedinfullwithanewproofvariation.
• AunifiedpresentationofthePartitionTheorem(currentlyspreadacrossseveralresearcharti-
cles)includingastrengthenedformthatimpliesandgeneralizesthesubsequencesumcaseof
theDeVos-Goddyn-MoharTheorem.AgeneralweightedversionvalidforR-modules(and
moregeneralhomomorphismweights)isalsopresented.
• TherecentpartialgeneralizationofPollard’sTheoremtoGeneralAbeliangroups,aswellas
relatedprogressbyHamidouneandSerra,isgivenwithimprovedbounds.
• The3k−4TheoremfordistinctsummandsinZisgiveninaslightlymoregeneralformand
also includes the recent complementary result concerning containment of long arithmetic
Preface vii
progressionsinthesumset.Amoregeneralversionofthetechniqueofmodularreductionis
included.
• The 3k−4 Theorem in C , with p prime, is given in several new variations in which the
p
logarithmic(resp.linear)restrictionisshiftedfromthecardinalitiesofthesetsontothe‘ad-
ditiveconstant’|A+B|−|A|−|B|+1.Inparticular,noassumptionontherelativesizeof
|A|versus|B|isrequired.
• Recentpreciseboundsforasumsetbeingisomorphictoatorsion-freesumsetaregivenasa
simpleconsequenceoftheboundonthetorsionsubgroupoftheuniversalambientgroup.
• Snevily’sConjecture,onlyrecentlyproven,isincluded.
• TheSavchev-ChenStructureresultforlongzero-sumfreesequencesisalsoincluded.
• Weightedsubsequencesumsaretreatedinthegeneralcontext,recentlyintroducedbyYuan
andZheng,ofusinghomomorphismsasweights.
• TheCombinatorialNullstellensatzisincludedalongwiththerecentlyprovenpuncturedver-
sion.
Thereis,ofcourse,muchmoreinthetext,butforthis,thereadershouldnowpressahead
into the main bulk of the text. Worth noting is that definitions for symbolic notation can be
foundviatheindex,thoughmostrecurringnotationisintroducedinChapters1,2and10,with
somenotableexceptionsinSection4.3(stabilizersandperiodicity),Section7.2(relativecom-
plements),andSection13.1(additionalsetpartitionnotation).Asregardssuggestionsforusing
thetextinacourse,thereareseveralpossibilities.Regardlessofwhatiscovered,itwillbehelp-
fultoreviewthenotationalconventionscontainedinthejustmentionedchapters/sections.PartI
canbeusedsimplyas anintroductorycourseinbasicstructuralsumsetresults.Alternatively,
thematerialfromPartIcanbetreatedmuchmorerapidlybypresentingonlythefinitesumset
casesofresults(thusskippingChapters3and4)andavoidingcertainpurposelyplacedredun-
dancies.Forinstance,theMultiplicityandPigeonholeBoundsaswellasVosper’sTheorem(the
bulkofChapters5and8)canbederivedasrespectiveconsequencesofKneser’sTheoremand
theKempermanStructureTheorem.Theyhavebeengivenseparateproofsbothtohighlightthe
importanceofthesespecialcasesandthemethodsusedthereaswellastograduallyaccustom
thestudenttothecomplexitiesofthemoregeneralresults.Forstudentsalreadyfamiliarwith
basic structural sumset results (either by prerequisite or during the second half of a course),
thechaptersfromPartsIIandIIIcancombinedinmostanyway.However,certainchaptersfit
wellinsuccession.Forinstance,Chapters10and13–16workwelltogetherasdoChapters7–8
and19–21.Infact,ifaninstructorissomewhatcarefulaboutwhichtopicsfromPartsIIandIII
arechosen,thenonlyaminimalamountofPartIneedbecoveredbeforehand(Kneser’sThe-
orem being rather indispensable, and Chapter 7 and Section 8.1 needed if the build up to the
3k−4TheoremforC istobefollowedthroughChapters19–21),givinganotherpossibility
p
for an introductory course using the material from this text. Alternatively, a course focussed
onFreimanHomomorphismscanbeconstructedfromChapters1–3,6–7,9and20,withextra
materialregardinghomomorphismsforinfinitesumsetsalsoinChapter4.
FundingforcompletingthisprojectcameprincipallyfromtheAustrianScienceFund,FWF
projectnumberP21576-N18.IwouldalsoliketothankAlfredGeroldingerforgoingthrough
(partsof)thebookandprovidinghelpfulsuggestionsandcomments.
Graz,Austria DavidJ.Grynkiewicz
Contents
1 AbelianGroupsandCharacterSums . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 BasicNotationandTerminology . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 AbelianGroupsandIndependence. . . . . . . . . . . . . . . . . . . . . . . 2
1.3 CharacterTheoryandDualGroups . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
PartI Sumsets
2 IntroductiontoSumsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Sumsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 InfiniteSummands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 MultiplicityofRepresentationinaSumset . . . . . . . . . . . . . . . . . . 15
2.4 X-ComponentDecompositions . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 ArithmeticProgressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 H-CosetDecompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.7 InductiononWell-OrderedSets . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8 FreimanHomomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 SimpleResultsforTorsion-FreeAbelianGroups . . . . . . . . . . . . . . . . . 25
3.1 FreimanIsomorphismsintoZ . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 ABasicLowerBoundforTorsion-FreeSumsets . . . . . . . . . . . . . . . 26
3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 BasicResultsforSumsetswithanInfiniteSummand . . . . . . . . . . . . . . . 29
4.1 InfiniteSummands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 CofiniteandSemi-cofiniteSubsets. . . . . . . . . . . . . . . . . . . . . . . 31
4.3 PeriodicSubsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.4 SumsetswithMultipleInfiniteSummands. . . . . . . . . . . . . . . . . . . 39
4.5 MoreGeneralNotionsofIsomorphism . . . . . . . . . . . . . . . . . . . . 42
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5 ThePigeonholeandMultiplicityBounds . . . . . . . . . . . . . . . . . . . . . . 57
5.1 ThePigeonholeandMultiplicityBounds . . . . . . . . . . . . . . . . . . . 57
5.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6 PeriodicSetsandKneser’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . 61
6.1 Kneser’sTheorem:StatementandConsequences . . . . . . . . . . . . . . . 61
6.2 Kneser’sTheorem:TheProof . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
ix
x Contents
7 Compression,Complementsandthe3k−4Theorem . . . . . . . . . . . . . . . 71
7.1 The3k−4Theorem:StatementandOverview . . . . . . . . . . . . . . . . 71
7.2 RelativeComplements,SaturatedSubsetsandDualPairs . . . . . . . . . . . 73
7.3 ABriefAside:TheDiscreteBrunn-MinkowskiTheoreminDimension2 . . 79
7.4 ModularReduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.5 Towardsthe3k−4Theorem:ContainmentbyArithmeticProgressions . . . 86
7.6 Towardsthe3k−4Theorem:LongArithmeticProgressionsintheSumset . 90
7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8 AdditiveEnergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.1 AdditiveEnergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.2 Vosper’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.3 SidonSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
9 Kemperman’sCriticalPairTheory . . . . . . . . . . . . . . . . . . . . . . . . . 111
9.1 ReductiontotheAperiodicCase . . . . . . . . . . . . . . . . . . . . . . . . 111
9.2 TheElementaryPairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
9.3 Quasi-periodicDecompositionsandtheRecursiveConstruction . . . . . . . 112
9.4 SpecialConsiderationsforUniqueExpressionElements . . . . . . . . . . . 114
9.5 SaturabilityofCriticalPairs . . . . . . . . . . . . . . . . . . . . . . . . . . 114
9.6 PartialConversestoQuasi-periodicLifting . . . . . . . . . . . . . . . . . . 115
9.7 TheKempermanStructureTheorem(KST) . . . . . . . . . . . . . . . . . . 118
9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
PartII SubsequenceSums
10 Zero-Sums,SetpartitionsandSubsequenceSums . . . . . . . . . . . . . . . . . 135
10.1 Zero-SumNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
10.2 ExistenceofSetpartitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
10.3 TheErdo˝s-Ginzburg-ZivTheoremandaBasicBoundfortheDavenport
Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
10.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
11 LongZero-SumFreeSequencesoverCyclicGroups . . . . . . . . . . . . . . . 145
11.1 AdditiveIsomorphismsforSubsequenceSums . . . . . . . . . . . . . . . . 145
11.2 AuxiliaryLemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
11.3 TheSavchev-ChenStructureTheorem. . . . . . . . . . . . . . . . . . . . . 150
11.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
12 Pollard’sTheoremforGeneralAbelianGroups . . . . . . . . . . . . . . . . . . 155
12.1 GeneralLowerBoundsfort-RepresentableSums . . . . . . . . . . . . . . . 155
12.2 StructuralResultsfort-RepresentableSums . . . . . . . . . . . . . . . . . 156
12.3 LessRestrictedBoundsfort-RepresentableSums . . . . . . . . . . . . . . 174
12.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
13 TheDeVos-Goddyn-MoharTheorem . . . . . . . . . . . . . . . . . . . . . . . . 181
13.1 AdditionalNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
13.2 ProofoftheDeVos-Goddyn-MoharTheorem . . . . . . . . . . . . . . . . . 182
13.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
14 ThePartitionTheoremI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
14.1 WeightedSubsequenceSums . . . . . . . . . . . . . . . . . . . . . . . . . 197
14.2 ThePartitionTheoremVersusDeVos-Goddyn-Mohar . . . . . . . . . . . . 198
14.3 ThePartitionTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Contents xi
14.4 ProofoftheUnweightedVersion . . . . . . . . . . . . . . . . . . . . . . . 201
14.5 ProofoftheWeightedVersion . . . . . . . . . . . . . . . . . . . . . . . . . 205
14.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
15 ThePartitionTheoremII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
15.1 TwoCorollariesofthePartitionTheorem . . . . . . . . . . . . . . . . . . . 229
∗
15.2 AGroupTheoreticLemmaAboutd (G) . . . . . . . . . . . . . . . . . . . 230
15.3 ProofofthePartitionCorollaries . . . . . . . . . . . . . . . . . . . . . . . 231
15.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
16 TheΨ-WeightedGaoTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
16.1 Ψ-WeightedSums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
16.2 UsefulLemmas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
16.3 TheDerivationoftheGlobalVersionfromtheLocal . . . . . . . . . . . . . 250
16.4 TheProofoftheLocalVersion . . . . . . . . . . . . . . . . . . . . . . . . 254
16.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
PartIII AdvancedMethods
17 GroupAlgebras:AnUpperBoundfortheDavenportConstant . . . . . . . . . 265
17.1 GroupAlgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
17.2 AGenericUpperBoundfortheDavenportConstant . . . . . . . . . . . . . 267
17.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
18 CharacterandLinearAlgebraicMethods:Snevily’sConjecture . . . . . . . . . 271
18.1 AProblemRegardingMatrixDeterminants . . . . . . . . . . . . . . . . . . 271
18.2 TheProofofSnevily’sConjecture . . . . . . . . . . . . . . . . . . . . . . . 275
18.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
19 CharacterSumandFourierAnalyticMethods:r-CriticalPairsI . . . . . . . . 279
19.1 CharacterSums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
19.2 APartial3k−4TheoremforPrimeOrderGroups:TheSymmetricCase . . 286
19.3 A Partial 3k−4 Theorem for Prime Order Groups: Near Equal Sized
Summands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
19.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
20 FreimanHomomorphismsRevisited . . . . . . . . . . . . . . . . . . . . . . . . 299
20.1 UniversalAmbientGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
20.2 BasicResultsfortheUniversalAmbientGroup . . . . . . . . . . . . . . . . 305
20.3 TheUniversalAmbientGroupBelowthe3k−4Bound . . . . . . . . . . . 314
20.4 AnUpperBoundforUniversalAmbientTorsion . . . . . . . . . . . . . . . 316
20.5 ConsequencesoftheUniversalAmbientTorsionBoundI . . . . . . . . . . 321
20.6 AQuickReviewofLatticeTheoryandtheGeometryofNumbers . . . . . . 322
20.7 ConsequencesoftheUniversalAmbientTorsionBoundII . . . . . . . . . . 326
20.8 TheUniversalAmbientGroupofaQuotient . . . . . . . . . . . . . . . . . 333
20.9 ASemigroup-AlgorithmicApproachtoFreimanHomomorphisms . . . . . . 337
20.10 TheUniversalAmbientGroupBelowtheCauchy-DavenportBound . . . . . 346
20.11 AbstractIsomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
20.12 RectificationinPrimeOrderGroups . . . . . . . . . . . . . . . . . . . . . . 361
20.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
21 TheIsoperimetricMethod:SidonSetsandr-CriticalPairsII . . . . . . . . . . 367
21.1 TheIsoperimetricMethod:BasicNotionsandProperties . . . . . . . . . . . 367
21.2 IteratedSumsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
21.3 SidonSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375