1 These notes were written to accompany a mini-course

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1
Thesenoteswerewrittentoaccompanyamini-courseattheEasterSchool
on‘DynamicsandAnalyticNumberTheory’heldatDurhamUniversity,U.K.,
inApril2014.
MultipleRecurrenceandFindingPatternsinDenseSets(‘theChapter’)was
firstpublishedbyCambridgeUniversityPressaspartofamulti-volumework
edited by Dzmitry Badziahin, Alexander Gorodnik and Norbert Peyerimhoff
entitled“DynamicsandAnalyticNumberTheory”(‘theVolume’). T
(cid:13)c intheChapter,TimAustin,2015
(cid:13)c intheVolume,CambridgeUniversityPress,2015
CambridgeUniversityPress’scatalogueentryfortheVolumecanbefound
F
atwww.cambridge.org/9781107552371.
NB: The copy of the Chapter as displayed on this website is a draft, pre-
publicationcopyonly.Thefinal,publishedversionoftheChaptercanbepur-
chased through Cambridge University Press and other standard distribution
channelsaspartofthewider,editedVolumAe.Thisdraftcopyismadeavailable
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R
D
Contents
T
F
1 Multiple Recurrence and Finding Patterns in Dense Sets T.
Austin page3
1.1 Szemere´di’sTheoremanditsrelatives 3
1.2 Multiplerecurrence A 6
1.3 Backgroundfromergodictheory 11
1.4 Multiplerecurrenceintermsofself-joinings 25
1.5 Weakmixing 34
1.6 Roth’sTheorem 42
1.7 Towardsconvergenceingeneral 49
R
1.8 Satedsystemsandpleasantextensions 53
1.9 Furtherreading 60
Index 69
D
1
Multiple Recurrence and Finding Patterns in
Dense Sets
TimAustin(CourantInstitute,NYU)
251MercerSt,NewYork,NY10012,U.S.A. T
Email:[email protected],URL:cims.nyu.edu/∼tim
F
Abstract Szemere´di’s Theorem asserts that any positive-density subset of
theintegersmustcontainarbitrarilylongarithmeticprogressions.Itisoneof
thecentralresultsofadditivecombinatorics.AfterSzemeredi’soriginalcom-
binatorial proof, Furstenberg noticed theAequivalence of this result to a new
phenomenon in ergodic theory that he called ‘multiple recurrence’. Fursten-
berg then developed some quite general structural results about probability-
preservingsystemstoprovetheMultipleRecurrenceTheoremdirectly.Fursten-
berg’sideashavesincegivenrisetoalargebodyofworkaroundmultiplere-
currenceandtheassociated‘non-conventional’ergodicaverages,andtofurther
R
connectionswithadditivecombinatorics.
This course is an introduction to multiple recurrence and some of the er-
godic theoretic structure that lies behind it. We begin by explaining the cor-
respondence observed by Furstenberg, and then give an introduction to the
necessarybackgroundfromergodictheory.Weemphasizetheformulationof
multipDlerecurrenceintermsofjoiningsofprobability-preservingsystems.The
nextstepisaproofofRoth’sTheorem(thefirstnontrivialcaseofSzemeredi’s
Theorem), which illustrates the general approach. We finish with a proof of
a more recent convergence theorem for some non-conventional ergodic aver-
ages,showingsomeofthenewerideasinuseinthisarea.
Theclassicintroductiontothisareaofcombinatoricsandergodictheoryis
Furstenberg’sbook[Fur81],butthetreatmentbelowhasamoremodernpoint
ofview.
1.1 Szemere´di’sTheoremanditsrelatives
In1927,vanderWaerdengaveaclevercombinatorialproofofthefollowing
surprisingfact:
4 T.Austin
Theorem1.1(VanderWaerden’sTheorem[vdW27]) Foranyfixedintegers
c,k ≥1, if the elements of Z are coloured using c colours, then there is a
nontrivialk-termarithmeticprogressionwhichismonochromatic:thatis,there
aresomea∈Zandn≥1suchthat
a,a+n,...,a+(k−1)n.
allhavethesamecolour. T
ThisresultnowfitsintoawholeareaofcombinatoricscalledRamseyThe-
ory.Theclassicaccountofthistheoryisthebook[GRS90].
Itiscrucialtoallowboththestartpointaandthecommondifferencen≥1
tobechosenfreely.Thistheoremhassomemoredifficultrelativeswhichallow
F
certainrestrictionsonthechoiceofn,butifonetriestofixasinglevalueofn
apriorithentheconclusioniscertainlyfalse.
In1936,Erdo˝sandTura´nrealizedthatadeeperphenomenonmightliebe-
neathvanderWaerden’sTheorem.Observethatforanyc-colouringofZand
foranyfinitesubintervalofZ,atleastoneoAfthecolour-classesmustoccupy
at least a fraction 1/c of the points in that subinterval. In [ET36] they asked
whetheranysubsetofZwhichhas‘positivedensity’inarbitrarilylongsubin-
tervalsmustcontainarithmeticprogressionsofanyfinitelength.
Thisturnsouttobetrue.Theformalstatementrequiresthefollowingdefi-
nition.WegiveitforsubseRtsofZd,d≥1,forthesakeofacominggeneraliza-
tion.Let[N]:={1,2,...,N}.
Definition1.2(UpperBanachdensity) ForE ⊆Zd,itsupperBanachden-
sityisthequantity
|E∩(v+[N]d)|
D d¯(E):=limsupsup .
Nd
N−→∞ v∈Zd
Thatis,d¯(E)isthesupremumofthoseδ >0suchthatonecanfindcubical
boxesinZdwitharbitrarilylongsidessuchthatEcontainsatleastaproportion
δ ofthelatticepointsinthoseboxes.
Exercise ProvethatDefinition1.2isequivalentto
d¯(E)=limsup sup(cid:110)(cid:12)(cid:12)E∩∏di=1[Mi,Ni](cid:12)(cid:12) : N ≥M +L∀i=1,2,...,d(cid:111).
L−→∞ ∏di=1(Ni−Mi) i i
(cid:67)
Theorem 1.3 (Szemere´di’s Theorem) If E ⊆Z has d¯(E)>0, then for any
k≥1therearea∈Zandn≥1suchthat
{a,a+n,...,a+(k−1)n}⊆E.
MultipleRecurrenceandFindingPatternsinDenseSets 5
Thespecialcasek=3ofthistheoremwasprovedbyRothin[Rot53],soit
is called Roth’s Theorem. The full theorem was finally proved by Szemere´di
in[Sze75].
Asalreadyremarked,Szemere´di’sTheoremimpliesvanderWaerden’sThe-
orem,becauseifZiscolouredusingccoloursthenatleastoneofthecolour-
classesmusthaveupperBanachdensityatleast1/c.
Szemere´di’s proof of Theorem 1.3 is one of the virtuoso feats of Tmodern
combinatorics. It is also the earliest major application of several tools that
have since become workhorses of that area, particularly the Szemere´di Reg-
ularityLemmaingraphtheory.However,shortlyafterSzemere´di’sproofap-
peared, Furstenberg gave a new and very different proof using ergodic the-
F
ory. In [Fur77], he showed the equivalence of Szemere´di’s Theorem to an
ergodic-theoreticphenomenoncalled‘multiplerecurrence’,andprovedsome
new structural results in ergodic theory which imply that multiple recurrence
alwaysoccurs.
MultiplerecurrenceisintroducedinthAenextsubsection.Firstwebringthe
combinatorialsideofthestoryclosertothepresent.FurstenbergandKatznel-
son quickly realized that Furstenberg’s ergodic-theoretic proof could be con-
siderablygeneralized,andin[FK78]theyobtainedamultidimensionalversion
ofSzemere´di’sTheoremasaconsequence:
Theorem 1.4 (FurstenRberg-Katznelson Theorem) If E ⊆Zd has d¯(E)>0,
andife ,...,e ,isthestandardbasisinZd,thentherearesomea∈Zd and
1 d
n≥1suchthat
{a+ne ,...,a+ne }⊆E
1 d
(so‘deDnsesubsetscontainthesetofouterverticesofanuprightright-angled
isoscelessimplex’).
This easily implies Szemere´di’s Theorem, because if k ≥ 1, E ⊆ Z has
d¯(E)>0,andwedefine
Π:Zk−1−→Z:(a ,a ,...,a )(cid:55)→a +2a +···+(k−1)a ,
1 2 k−1 1 2 k−1
thenthepre-imageΠ−1(E)hasd¯(Π−1(E))>0,andanuprightisoscelessim-
plexfoundinΠ−1(E)projectsunderΠtoak-termprogressioninE.Similarly,
byprojectingfromhigher-dimensionstolower,onecanprovethatTheorem1.4
actuallyimpliesthefollowing:
Corollary 1.5 If F ⊂Zd is finite and E ⊆Zd has d¯(E)>0, then there are
somea∈Zd andn≥1suchthat{a+nb: b∈F}⊆E.
6 T.Austin
Forabouttwentyyears,theergodic-theoreticproofofFurstenbergandKatznel-
sonwastheonlyknownproofofTheorem1.4.Thatchangedwhenanewap-
proach using hypergraph theory was developed roughly in parallel by Gow-
ers [Gow06], Nagle, Ro¨dl and Schacht [NRS06] and Tao [Tao06b]. These
worksgavethefirst‘finitary’proofsofthistheorem,implyingsomewhateffec-
tive bounds: unlike the ergodic-theoretic approach, the hypergraph approach
gives an explicit value N =N(δ) such that any subset of [N]d containinTg at
least δNd points must contain a whole simplex. (In principle, one could ex-
tract such a bound from the Furstenberg-Katznelson proof, but it would be
extremelypoor:seeTao[Tao06a]fortheone-dimensionalcase.)
The success of Furstenberg and Katznelson’s approach gave rise to a new
F
sub-fieldofergodictheorysometimescalled‘ergodicRamseytheory’.Itnow
contains several other results asserting that positive-density subsets of some
kind of combinatorial structure must contain a copy of some special pattern.
Someofthesehavebeenre-provenbypurelycombinatorialmeansonlyvery
recently.WewillnotstatetheseindetailhereA,butonlymentionbynametheIP
Szemere´diTheoremof[FK85],theDensityHales-JewettTheoremof[FK91]
(finallygivenapurelycombinatorialproofbythemembersofthe‘Polymath
1’ project in [Pol09]), the Polynomial Szemere´di Theorem of Bergelson and
Leibman[BL96]andtheNilpotentSzemere´diTheoremofLeibman[Lei98].
R
1.2 Multiplerecurrence
1.2.1 Thesettingofergodictheory
Ergodic theory studies the ‘statistical’ properties of dynamical systems. The
D
following treatment is fairly self-contained, but does assume some standard
facts from functional analysis and probability, at the level of advanced text-
bookssuchasFolland[Fol99]orRoyden[Roy88]andBillingsley[Bil95].
Let G be a countable group; later we will focus on Z or Zd. A G-space
is a pair (X,T) in which X is a compact metrizable topological space, and
T =(Tg) is an action of G on X by Borel measurable transformations:
g∈G
thus,
Te=id and Tg◦Th=Tgh ∀g,h∈G,
X
whereeistheidentityofG.AZ-actionT isspecifiedbythesingletransforma-
tionT1whichgeneratesit.Similarly,aZd-actionT maybeidentifiedwiththe
commuting d-tuple of transformations Tei, where e1, ..., ed are the standard
basisvectorsofZd.
The set of Borel probability measures on a compact metrizable space X is
MultipleRecurrenceandFindingPatternsinDenseSets 7
denotedbyPr(X).Thissetiscompactintheweak(cid:63)topology:forinstance,this
can be seen by the Riesz Representation Theorem ([Fol99, Theorem 7.17]),
whichidentifiesPr(X)withaclosed,bounded,convexsubsetoftheBanach-
spacedualC(X)(cid:63),followedbyanapplicationofAlaoglu’sTheorem([Fol99,
Theorem5.18]).
Next,let(X,S)and(Y,T)beanytwomeasurablespaces,letµ beaprob-
ability measure on S, and let π :X −→Y be measurable. Then theTimage
measureofµ underπ isthemeasureπ µ onT definedbysetting
∗
(π µ)(B):=µ(π−1(B)) ∀B∈T.
∗
If X andY are compact metrizable spaces, then π defines a map Pr(X)−→
∗
F
Pr(Y). If, in addition, π is continuous, then π is continuous for the weak(cid:63)
∗
topologies.
Finally, a G-system is a triple (X,µ,T) in which (X,T) is a G-space and
µ isaT-invariantmemberofPr(X),meaningthatTgµ =µ foreveryg∈G.
∗
When needed, the Borel σ-algebra of XAwill be denoted by B . We often
X
denoteaG-system(X,µ,T)byasingleboldfacelettersuchasX.AZ-system
willsometimesbecalledjustasystem.
The definitions above ignore a host of other possibilities, such as dynam-
icswithaninfiniteinvariantmeasure,orwithanon-invertibletransformation.
Ergodictheoryhasbranchesforthesetoo,buttheydonotappearinthiscourse.
R
Examples
1. LetX =T=R/Zwithitsusualtopology,letµ beLebesguemeasure,and
letT betherotationbyafixedelementα ∈X:
D Tx:=x+α.
Thisiscalledacirclerotation.
2. Letp=(p ,...,p )beastochasticvector:thatis,aprobabilitydistribution
1 m
ontheset{1,2,...,m}.LetX :={1,2,...,m}Z withtheproducttopology,
letµ:=p⊗Z(thelawofani.i.d.randomsequenceofnumberseachchosen
accordingtop),andletT betheleftwardcoordinate-shift:
T((x ) ):=(x ) .
n n n+1 n
ThisiscalledtheBernoullishiftoverp. (cid:67)
AnimportantsubtletyconcernsthetopologyonX.Inmostofergodicthe-
ory, no particular compact topology on X is very important, except through
theresultingBorelσ-algebra:itisthismeasurablestructurethatunderliesthe
theory. This is why we allow arbitrary Borel measurable transformations Tg,
8 T.Austin
ratherthanjusthomeomorphisms.However,generalmeasurablespacescanex-
hibitcertainpathologieswhichBorelσ-algebrasofcompactmetrizablespaces
cannot. The real assumption we need here is that our measurable spaces be
‘standardBorel’,buttheassumptionofacompactmetricisaconvenientway
toguaranteethis.
Havingexplainedthis,bewarethatmanyauthorsrestricttheconvenientterm
‘G-space’toactionsbyhomeomorphisms. T
1.2.2 Thephenomenonofmultiplerecurrence
Inordertointroducemultiplerecurrence,itishelpfultostartwiththeprobability-
F
preservingversionofPoincare´’sclassicalRecurrenceTheorem.
Theorem1.6(Poincare´Recurrence) If(X,µ,T)isasystemandA∈B has
X
µ(A)>0,thenthereissomen(cid:54)=0suchthatµ(A∩T−nA)>0.
Proof The pre-images T−nA are all subseAts of the probability space X of
equalpositivemeasure,sosometwoofthemmustoverlapinpositivemeasure.
Oncewehaveµ(T−nA∩T−mA)>0forsomen(cid:54)=m,theinvarianceofµunder
Tnimpliesthatalsoµ(A∩Tn−mA)>0.
Furstenberg’smainresultfrom[Fur77]strengthensthisconclusion.Heshows
thatinfactonemayfindsReveralofthesetsT−nA,n∈Z,thatsimultaneously
overlapinapositive-measureset,wheretherelevanttimesnformanarithmetic
progression.
Theorem 1.7 (Multiple Recurrence Theorem) If (X,µ,T) is a system and
A∈B hasµ(A)>0,thenforanyk≥1thereissomen≥1suchthat
X
D
µ(T−nA∩···∩T−knA)>0.
TheMultidimensionalMultipleRecurrenceTheoremfrom[FK78]provides
ananalogofthisforseveralcommutingtransformations.
Theorem1.8(MultidimensionalMultipleRecurrenceTheorem) If(X,µ,T)
isaZd-systemandA∈B hasµ(A)>0,thenthereissomen≥1suchthat
X
µ(T−ne1A∩···∩T−nedA)>0.
Notethatford=2,simplyapplyingthePoincare´ RecurrenceTheoremfor
thetransformationTe1−e2 givestheconclusionofTheorem1.8.
This course will include a proof of Theorem 1.7 in the first case beyond
Poincare´ Recurrence:k=3.Twodifferentergodic-theoreticproofsofthefull
Theorem1.8canbefoundin[Fur81]and[Aus10c].Thesearetoolongtobe
MultipleRecurrenceandFindingPatternsinDenseSets 9
includedinthiscourse,butwewillformulateandprovearelatedconvergence
resultwhichgivesanintroductiontosomeoftheideas.
First, let us prove the equivalence of Theorem 1.8 and Theorem 1.4. This
equivalence is often called the ‘Furstenberg correspondence principle’. Al-
thougheasytoprove,ithasturnedouttobeahugelyfruitfulinsightintothe
relationbetweenergodictheoryandcombinatorics.Theversionwegivehere
essentiallyfollows[Ber87]. T
Proposition1.9(Furstenbergcorrespondenceprinciple) IfE⊆Zd,thenthere
areaZd-system(X,µ,T)andasetA∈B suchthatµ(A)=d¯(E),andsuch
X
thatforanyv ,v ,...,v ∈Zd onehas
1 2 k
d¯((E−v )∩(E−v )∩···∩(E−v ))≥µ(T−v1FA∩···∩T−vkA).
1 2 k
Inordertovisualizethis,observethat
(E−v )∩(E−v )∩···∩(E−v )
1 2 k
A
isthesetofthosea∈Zd suchthata+v ∈E foreachi≤k.Itsdensitymaybe
i
seen as the ‘density of the set of translates of the pattern {v ,v ,...,v } that
1 2 k
lie entirely inside E’. In these terms, the above propositions asserts that one
cansynthesizeaZd-systemwhichprovidesalowerboundonthisdensityfor
anygivenpatternintermsoftheintersectionofthecorrespondingshiftsofthe
subsetA. R
Proof ChooseasequenceofboxesRj:=∏di=1[Mj,i,Nj,i]suchthat
min (N −M )−→∞ as j−→∞
j,i j,i
i∈{1,2,...,d}
and D
|E∩R |
j −→d¯(E) as j−→∞.
|R |
j
We can regard the set E as a point in the space X :=P(Zd) of subsets of
Zd, on which Zd naturally acts by translation: TnB:=B−n. This X can be
identifiedwiththeCartesianproduct{0,1}Zd byassociatingtoeachsubsetits
indicatorfunction.Itthereforecarriesacompactmetrizableproducttopology
whichmakes(X,T)aZd-space.
Let
1
νj:= |R | ∑ δTn(E) foreach j,
j n∈Rj
theuniformmeasureonthepieceoftheT-orbitofE indexedbythelargebox
R .Becausetheside-lengthsoftheseboxesalltendto∞,thesemeasuresare
j
10 T.Austin
approximately invariant: that is, (cid:107)T−mν −ν (cid:107) −→0 as j −→∞ for any
∗ j j TV
fixed m∈Zd, where (cid:107)·(cid:107) is the total variation norm (see [Fol99, Section
TV
7.3]).
Since Pr(X) is weak(cid:63) compact, we may let µ ∈Pr(X) be a subsequential
weak(cid:63) limitofthemeasuresν .Bypassingtoasubsequence,wemayassume
j
that in fact ν −→ µ (weak(cid:63)). Since the measures ν are approximately T-
j j
invariant,andeachT−mactscontinuouslyfortheweak(cid:63)topologyonPr(XT),µ
∗
itselfisstrictlyT-invariant.
Finally, let A:={H ∈X : H (cid:51)0}. This corresponds to the cylinder set
{(ω ) : ω =1}⊆{0,1}Zd. We will show that (X,µ,T) and A have the
n n 0
desiredproperties.ByourinitialchoiceofthesequenceofboxesR ,wehave
j
F
1 |E∩R |
µ(A)= j−li→m∞νj(A)= j−li→m∞|Rj|n∑∈Rj1Tn(E)(0)= j−li→m∞ |Rj|j =d¯(E).
The first convergence here holds because 1 is a continuous function for the
A
producttopologyonX,andsoweak(cid:63)converAgenceappliestoit.
Ontheotherhand,foranyv ,v ,...,v ∈Zd,theindicatorfunction1
1 2 k T−v1A∩···∩T−vkA
isalsocontinuousonX,andso
µ(T−v1A∩···∩T−vkA)= lim νj(T−v1A∩···∩T−vkA)
j−→∞
1
R= j−li→m∞|Rj|n∑∈Rj1Tn+v1E∩···∩Tn+vkE(0)
1
= j−li→m∞|Rj|n∑∈Rj1Tv1E∩···∩TvkE(n)
≤d¯(Tv1E∩···∩TvkE),
D
sincetheupperBanachdensityisdefinedbyalimsupoverallbox-sequences
withincreasingside-lengths.
Corollary1.10 Theorems1.4and1.8areequivalent.
Proof (=⇒) Let(X,µ,T)beaZd-systemandA∈B withµ(A)>0.For
X
eachx∈X letE :={n∈Zd : Tnx∈A},andforeachN∈Nlet
x
Y :={x∈X : |E ∩[N]d|≥µ(A)Nd/2}.
N x
Asimplecalculationgives
(cid:90)
µ(Y )Nd+µ(X\Y )µ(A)Nd/2≥ |E ∩[N]d|µ(dx)
N N x
X
(cid:90) (cid:90)
= ∑ |Ex∩{n}|µ(dx)= ∑ 1T−n(A)dµ =µ(A)Nd,
n∈[N]d X n∈[N]d X