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Higher order fourier analysis as an algebraic theory I.

BAL AZS SZEGEDY

October 25, 2018

Abstract

Ergodic theory, Higher order fourier analysis and the hypergraph regularity method are three

possible approaches to Szemeredi type theorems in abeliangroups. In this paper we develop an

algebraic theory that creates a connection between these approaches. Our main method is to take the

ultra product of abelian groups and to develop a precise algebraic theory of higher order characters

on it. These results then can be turned back into approximative statements about finite Abelian

groups.

Contents

1 Introduction 2

1.1 Explanation of the Results and further remarks . . . . . . . .. . . . . . . . . . . 3

1.2 Characteristic Factors . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 4

2 Manipulating with σ-algebras 5

2.1 Basics and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 5

2.2 Measure andσ-topology on ultra-products . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Perfectσ-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 σ-algebras in hyper-graph regularization and Gowers’s norms . . . . . . . . . . . . 6

2.5 Weak Orthogonality and cosetσ-algebras . . . . . . . . . . . . . . . . . . . . . . 7

2.6 Modules and relative separability . . . . . . . . . . . . . . . . . .. . . . . . . . . 8

2.7 Slices of measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 9

2.8 Relative separable elements and Fourierσ-algebras . . . . . . . . . . . . . . . . . 11

2.9 Characterization of the Fourierσ-algebras . . . . . . . . . . . . . . . . . . . . . . 12

2.10 Group extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 13

2.11 Cubic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 15

2.12 Face actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 17

1

2.13 Isometric extensions are Abelian . . . . . . . . . . . . . . . . . .. . . . . . . . . 18

3 Higher order Fourier analysis 18

3.1 Definition of higher order Fourier analysis . . . . . . . . . . .. . . . . . . . . . . 18

3.2 Higher order group algebras . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 19

3.3 Decomposition of the elements inCk . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Proof of the fundamental theorem . . . . . . . . . . . . . . . . . . . .. . . . . . 21

1 Introduction

This paper is an announcement of a new approach to higher order Fourier analysis. It contains proofs

and some explanations but a longer, more detailed version ison the way. The next paper will contain

further results on the structure of higher order charactersand will have a special emphasis on the

finite translations of our results. Note that every result onultra products in this paper automatically

translates to an approximative finite result with epsilons.

In a paper [1] by G. Elek and myself we develop a measure theoretic approach to dense hyper-

graphs. The central theorem of the paper creates a correspondence between growing sequences of

hypergraphs and various measurable sets. A powerful feature of that approach is that extra alge-

braic structures (such as groups structures) on the hypergraphs can be detected on the limit objects

and thus can be related to the regularity lemma. The general reason is that ultra products preserve

axiomatizable structures.

Our starting point is the following question. LetS be a subset of a finite abelian groupA and let

H be thek uniform hypergraph consisting of thek-edges

{(a1, a2, . . . , ak) | a1 + a2 + . . .+ ak ∈ S}.

What is the regularization ofH in terms of the structure ofS?

More generally, iff : A→ C is a bounded function, how can we regularizef(x1+x2 + . . .+xk)?

In the paper [1] we prove a certain uniqueness of regularization. This suggest that there must be

a nice algebraic answer to the above problem. It turns out that for k = 2 ordinary Fourier analysis of

f gives a full answer to the question (when dominant terms in the Fourier expansion are considered).

However fork > 2 a higher order version of Fourier analysis is needed. Such a theory was started

first by Gowers in [3] and later continued by Green, Host, Kra,Tao, Ziegler and many others (see

reference list).

Our contribution to the subject is a clean algebraic versionof Higher order Fourier analysis (on

the ultra product of finite abelian groups) which is quite similar to ordinary Fourier analysis. For

everyn there are ordern characters on our ultra product Abelian group that form an orthonormal

basis for certain Hilbert spaces depending onn. These characters can be described as generators

2

of rank-one modules. Further more then-th order characters are forming an Abelian group with

respect to point wise multiplication that we call then-th dual group.

We emphasize that on a finite Abelian group ordinary linear characters form a basis of the dual

space and this phenomenon leaves no room for higher order characters that are fully orthogonal to

the ordinary ones. However on the ultra product group the first order characters are not forming a

full basis. This allows us to develop our algebraic theory.

Note that our results can be extended to compact Abelian groups but we will do it in a forthcom-

ing paper. Also the relationship between higher order characters and nil-systems will be discussed

in a the second part of this paper.

Since translating the results back to finite statements is a routine thing we don’t focus on it. We

give the details in a longer version of the same paper.

Finally we mention that our language allows us to define a version of higher order representation

theory for non-commutative groups but we don’t know applications yet.

1.1 Explanation of the Results and further remarks

Is this section we give a short explanation of the results in the paper.

Let {Ai}∞i=1 be an increasing sequence of finite Abelian groups and letA be the ultra product of

them. Using the language from [1] there is aσ-algebraA and a shift invariant probability measure

µ onA.

Let λi : Ai → C a sequence of linear characters. It is easy to see that the ultra limit of these

functions is a linear character onA. Functions arising this way bill be calledfirst order characters

on A. It turns out that first order characters are not forming a basis in L2(A,A, µ). There is a

smaller sigma algebraF1 such that they are an orthonormal basis inL2(A,A, µ). We callF1 the

first order Fourier σ-algebra.

A result in this paper gives a simple characterization for the elements inF1. A measurable setM

is inF1 if and only if theσ-algebra generated by the shiftsM +x ofM is separable. In other words

countable many shifts ofM generate all the shifts. We call such setsM separable elementsin A.

Note thatF1 is not a separableσ-algebra despite of the fact that every element of it is separable.

Thesecond order Fourier analysiscomes from an interesting phenomenon. The are measur-

able setsM in A that are not separable but countable many shifts ofM together withF1 generate

all the shifts ofM . Such elements are calledrelative separableelements with respect toF1. These

elements are forming aσ-algebra which is denoted byF2.

Continuing this processFn denotes theσ-algebra of relative separable elements with respect to

Fn−1. We say thatFn is then-th order Fourierσ-algebra.

Our definition ofHigher order Forier analysis is the following:DecomposeL2(Fn) into irre-

3

ducibleL∞(Fn−1) modules!

Themain theorem in this paper says thatL2(Fn) can uniquely be decomposed intorank one

L∞(Fn−1) modulesthat are pairwise orthogonal to each other. Every such module is generated by

a functionλ which takes complex values of absolute value one. These functions are measurable in

Fn and furthermoref(x) := λ(x)λ(x + t) is measurable inFn−1. The functionλ us unique up to

multiplication with elements fromL∞(Fn−1). Note thatF0 is defined to be the trivialσ-algebra.

This shows that the first order Fourier analysis gives back the classical theory.

Our language is basicallyergodic theoretic. One can look atA as a measure preserving system

acting on itself. Since theσ-algebrasFi are shift invariant, they can be regarded asfactors of the

system. Our main theorem says in this formulation thatFn is an Abelian extension ofFn−1. A lots

of intuition for this paper (especially the part about cubicstructures) was borrowed from a beautiful

paper by Host and Kra [6].

In a continuation of this paper we will analyze these systemsmore with a special emphasis on

their relationship with nil-systems.

We mention that our theory can be looked at from thehypergraph regularization point of view:

If f : A → C is a bounded function then the “first level” regularization of then-variable function

f(x1, x2, . . . , xn) is obtained by taking the projectiong = E(f |Fn−1) and then consideringg(x1+

x2 + . . .+ xn).

The so calledGowers’s normUn(f) is 0 if and only if f is orthogonal toFn−1. In other words

for every bounded functionf : A → C there is a (unique!) decomposition

f = g + f1 + f2 + f3 + . . .

converging inL2 such thatUn(g) = 0 and the functionsfi are contained in different rank-one

modules overL∞(Fn−2).

We mention that a part of this theory can be developed for non-commutative groups. One can

take their ultra product and then consider that tower of relative separableσ-algebra. It turns out that

they areisometric extensionsof each other. Details will be worked out later.

1.2 Characteristic Factors

In this short chapter we explain an interesting phenomenon.The concept of acharacteristic factor

is crucial in all the three theories: Ergodic theory, Higherorder Fourier analysis and Hypergraph

regularization. Later on, when we relate the three theoriesit will be apparent.

Characteristic factors are classical in Ergodic theory. Itturns out that certain averages in measure

preserving systems remain the same when the set gets projected to a courserσ-algebra. The smallest

such algebra is characteristic for the type of average.

4

In hypergraph theory they show up in the following way. The socalled counting lemma in the

infinite setting [1] says that if we want to count sub-hypergraphs in ak-uniform hypergraph then

it is enough to project the the indicator function of the edgeset to certain courserσ-algebras. The

coursest suchσ-algebra gives rise to hypergraph regularization.

As it is pointed out in the present paper, Higher order Fourier analysis can also be interpreted in

a similar way. There we project functions to the Fourierσ-algebras.

2 Manipulating with σ-algebras

2.1 Basics and notation

Throughout this paper we fix a non-principal ultra-filterω on the natural numbersN. For a sequence

of sets{Xi}∞i=1 we denote their ultra product by[{Xi}∞i=1]. We will also use the shorthand notation

[{Xi}] once the running indexi is clear from the context. For a bounded sequencea1, a2, . . . of

complex numbers we denote their ultra limit bylimω ai.

2.2 Measure andσ-topology on ultra-products

Let ω be a fixed ultrafilter onN. Let {Xi}∞i=1 be an infinite sequence of finite sets. We denote their

ultra product[{Xi}∞i=1] byX.

Definition 2.1 A subsetH ⊆ X is said to beperfect if

H = [{Hi}∞i=1]

for some subsetsHi ⊆ Xi. A subsetH ⊆ X is said to beopenif it is the union of at most countable

many perfect sets.

We denote byP(X) the collection of perfect sets and byO(X) the open sets onX. We say that

the open sets define aσ-topologyonX. ForH ∈ P(X) we introduce the measureµ(X) as

µ(H) = limω(|Hi|/|Xi|).

The sigma algebra generated byP(X) onX will be denoted byA = A(X). The measureµ extends

to aσ-additive probability measure onA.

Lemma 2.1 (countable compactness)If A is the union of countable many open sets{Hi}∞i=1 then

there is a natural numbern such thatA = ∪ni=1Hi.

Definition 2.2 A functionf : X → C is calledperfect if

f([xi∞i=1]) = lim

ωfi(xi)

for some sequence of functionsfi : Xi → C.

5

2.3 Perfectσ-algebras

Let {Xi}∞i=1 be a sequence of finite sets and let{2Xi}∞i=1 be the sequence of their power sets. The

elements of the ultra product

[{2Xi}∞i=1]

are in a one to one correspondence withP(X) so by abusing the notation we can identify the two

objects. The advantage of this representation ofP(X) is that it creates aσ-topology onP(X).

Definition 2.3 The sigma algebraB ⊆ A(X) is called perfect if it is generated by a set in

P(P(X)).

An important advantage of perfectσ-algebras is that the conditional expectation operator be-

haves in a nice measurable way.

Definition 2.4 LetB ⊆ A be aσ-algebra. We say thatB has theprojection property if there is a

A(P(X)×X)-measurable function

f : P(X)×X → [0, 1]

such that for everyH ∈ P(X)

f(H, x) = E(H|B)(x).

Lemma 2.2 Any perfectσ-algebra has the projection property.

Proof. Assume thatB is generated by the ultra product of the sequence{Si}∞i=1 whereSi ⊆ 2Xi .

Let S be the ultra product of the sequence{Si}∞i=1. For every natural numbern andi we construct

functionsfn,i : 2Xi × Xi → [0, 1] in the following way. Let us fix a subsetH ⊆ Xi and take

the projections ofH to all finite σ-algebras generated by at mostn elements fromSi. Each such

projection is a function of the formXi → [0, 1]. We define the functionx 7→ fn,i(H,x) as one

of the projections that has minimalL2 norm among all possible projections. We denote byfn the

ultra limit function of the sequence{fn,i}∞i=1. One can observe that for every fixed perfect setH

the functionx → fn(H, x) is a projection ofH to a finiteσ-algebra generated byn-sets from

S. Furthermore its norm is minimal among all such projections. This implies that the measurable

functionlimn→∞ fn has the desired property.

2.4 σ-algebras in hyper-graph regularization and Gowers’s norms

We review some of the language developed in [1] to regularizehypergraphs with ultra product. Let

k be a fixed natural number andS be a subset of{1, 2, 3, . . . , k}. We denote byPS the natural pro-

jection fromXk to X

S . The pre-image ofA(XS) underXk will be denoted byσ(S). Furthermore

6

σ(S)∗ denotes theσ-algebra generated by all theσ-algebrasσ(S′) whereS′ is a proper non-empty

subset ofS.

A setH ∈ A(Xk) is calledquasi-random if the projectionE(H|σ([k])∗) is the constant func-

tion. Letf : Xk → C be anL2 function. The functionf is called quasi-random ifE(f |σ([k]∗) is the

zero function. The nice thing about quasirandom functions is that they are forming a Hilbertspace.

Gowers ... introduced the so calledoctahedral-normsthe characterize quasirandom finctions in the

finite setting. A similar lemma applies in the infinite setting but the epsilon’s are disappears.

Definition 2.5 The octahedral normOk(f) is the2k-th root of the integral

∫

xi,j∈X

∏

c∈{0,1}k

f(x1,c1 , x2,c2 , . . . , xk,ck)q(c)

where1 ≤ ı ≤ k , j ∈ {0, 1} andq(c) : C → C denotes the conjugation operator raised to the

power∑k

i=1 ci.

Lemma 2.3 The functionf : X → C is quasi-random if and only ifOk(f) = 0.

Definition 2.6 (Gowers’s norms) Let f : X → C be anL2-function and letfk : Xk → C denote

the functionfk(x1, x2, . . . , xk) = f(x1 + x2 + . . .+ xk). Then the Gowers normUk(f) is defined

asOk(fk).

2.5 Weak Orthogonality and cosetσ-algebras

LetH ⊆ L2(X, µ) be a Hilbert space of functions onX. We denote byPH the orthogonal projection

toH . We will need the following lemma

Lemma 2.4 (Integration in Hilbert spaces) Let f : X × Y → C be anA(X × Y) measurable

function such thatgy(x) = f(x, y) is in a Hilbert spaceH for everyy ∈ Y. Then

g(x) =

∫

y

f(x, y) dµ2

is inH .

Proof. Following the next equations

‖g(x)‖2 =

∫

x

∫

y

f(x, y)g(x) =

∫

y

∫

x

f(x, y)g(x) =

=

∫

y

∫

x

f(x, y)PH(g)(x) =

∫

y

∫

x

f(x, y)PH(g)(x) =

=

∫

x

g(x)PH(g)(x) = ‖PH(g)‖2

we get thatg = PH(g) which shows thatg ∈ H .

7

Definition 2.7 Two Hilbert spacesH1 andH2 in a bigger Hilbert spaceH will be calledweakly

orthogonal if for anyf ∈ H1 we have thatPH2(f) ∈ H1. LetB1 andB2 be two subσ-algebras in

the same measure space. We say thatB1 is weakly orthogonal toB2 if for anyf ∈ B1 the function

E(f |B2) is measurable inB1.

Lemma 2.5 Weak orthogonality is a symmetric relation.

Proof. Weak orthogonality for Hilbert spaces is clearly a symmetric notion. It means that the

orthogonal complements ofH1 ∩ H2 in H1 andH2 are orthogonal. Weak orthogonality of theσ-

algebrasB1 andB2 is equivalent with the weak orthogonality ofL2(B1) andL2(B2) so it is again

symmetric.

Definition 2.8 Let {Ai}∞i=1 be an infinite sequence of finite Abelian groups. We denote their ultra

product[{Ai}∞i=1] byA. A subgroupH ⊆ A is calledperfect if there are subgroupsHi ⊆ Ai with

H = [{Hi}∞i=1]. Thecosetσ-algebra corresponding to a perfect subgroupH is the subalgebra

A(A,H) ⊆ A consisting of the measurable sets which are unions of cosetsof H. A σ-algebra

B ⊆ A is calledA-invariant (or invariant) if for everyY ∈ B anda ∈ A we have thatY+ a ∈ B.

Clearly, any cosetσ-algebra is invariant. The advantage of cosetσ-algebras is that the projection

operator to them can be computed by an averaging operator. Let f : A → C be anL2 function.

Then we can define its averageT (f) according toH by

T (f)(x) =

∫

h∈H

f(x+ h).

Lemma 2.4 implies that iff is measurable in an invariantσ-algebraB thenT (f) is also measurable

in B. It is clear from the definition that(T (f), w) = (f, w) for everyw ∈ L2(A(A,H)) and that

T (f) ∈ L2(A(A,H)). We obtain thatT (f) coincides with the projectionE(f |A(A,H)). As a

consequence we get the next lemma.

Lemma 2.6 A cosetσ-algebra onA is weakly orthogonal to any invariantσ-algebra.

2.6 Modules and relative separability

Definition 2.9 Let B ⊆ A(X) be aσ-algebra and letR = L∞(B) be the ring of boundedB-

measurable functionsf : X 7→ C with the point wise multiplication. A Hilbert spaceH ⊆ L2(X)

is anR-module if hr ∈ H for everyh ∈ H andr ∈ R. Therank rk(H) is the minimal size of a

subsetS ⊆ H such that theR-module generated byS is dense inH .

Lemma 2.7 LetH1 andH2 be twoR modules. ThenPH2(vr) = rPH2

(v) wheneverv ∈ H1 and

r ∈ R.

8

Proof. We have that

(PH2(vr), w) = (vr, w) = (v, wr) = (PH2

(v), wr) = (rPH2(v), w)

holds for everyw ∈ H2. By settingw = PH2(vr) − rPH2

(v) we get that(w,w) = 0 which

completes the proof.

Lemma 2.8 LetH1 ⊆ H2 be twoR modules. Thenrk(H1) ≤ rk(H2).

Proof. Lemma 2.7 implies that ifS is a generating set ofH2 as anR module thenS′ =

{PH2(s) | s ∈ S} is a generator set ofH1.

Lemma 2.9 LetH1 ⊆ H2 beR-modules and letH3 be the orthogonal complement ofH1 in H2.

ThenH3 is anR-module andE(f g|B) is the0 function for everyf ∈ H1 andg ∈ H3.

Proof. We have that ifr ∈ R, h ∈ H3 andt ∈ H1 then(t, rh) = (tr, h) = 0 and sorh ∈ H3.

Let e = E(fg|B). We have that lemma 2.7 thatE(f ge|B)(x) = |e(x)|2 and using(f, ge) = 0 we

obtain that∫

|e(x)|2 = 0. This shows thate is the constant0 function.

Definition 2.10 LetB1,B2 ⊆ A(X) beσ-algebras. We say thatB2 is relative separableoverB1

if 〈B1,B2〉 can be generated byB1 and at most countable many extra elements as aσ-algebra.

If B1 ⊆ B2 then relative separability means thatL2(B2) is an at most countable rankL∞(B1)-

module.

Lemma 2.10 (Independent complement)LetB1 ⊆ B2 ⊆ A(X) be twoσ-algebras such thatB2

is relative separable and relative atom less overB1. Then there is a separableσ-algebraB3 ⊆ B2

which is independent fromB1 and〈B1,B3〉 = B2.

Lemma 2.11 (Relative basis)Let B1 ⊆ B2 ⊆ A(X) be twoσ-algebras such thatB2 is relative

separable and relative atom-less overB1. Then there is a system of at most countably many functions

fi : X → C such that

E(fifj |B1)(x) = δi,j .

Lemma 2.12 LetA(A,H) be a cosetσ-algebra,B be a subσ-algebra in it andB2 ⊆ A(A) a σ-

algebra such thatB is relative separable overB2. ThenB is relative separable overA(A,H)∩B2.

2.7 Slices of measurable sets

Let H ⊆ Xk be a subset and letf : Xk → {0, 1} be its characteristic function. In this section we

study the “slices” ofH according to the last coordinate. Letx be an element inX. Thex-slice of

H is the set

Hx := {(x1, x2, . . . , xk−1 | (x1, x2, . . . , xk−1, x) ∈ H}.

9

Theσ-algebra generated by allx-slices will be denoted byS(H).

Lemma 2.13 The setH is in σ([k])∗ if and only ifS(H) is relative separable overσ([k − 1])∗.

Proof. First we assume thatH ∈ σ([k])∗. Then there are countable many sets inAi,j ∈ σ([k]/{j})

with i ∈ N and1 ≤ j ≤ k such thatH is measurable in theσ-algebra generated by{Ai,j}. If j < k

thenS(Ai,j) ∈ σ([k − 1]∗) furthermoreS(Ai,k) is generated by one element. This shows that

S(H) ⊆ 〈S(Ai,j) | i ∈ N, 1 ≤ j ≤ k〉

is relative separable overσ([k − 1])∗.

For the other direction letB be a sigma algebra containingS(H) such thatB is relative separable

and relative atom less overσ([k − 1])∗. Such aσ-algebra can be constructed by extendingS(H)

by at most countable many generators. Letg1, g2, . . . be a relative basis ofB overσ([k − 1])∗. We

have that

f(x1, x2, . . . , xk) =∑

i

gi(z)E(fx(z)gi(z)|σ([k − 1])∗)

wherez = (x1, x2, . . . , xk−1). This formula show thatf is measurable inσ([k])∗.

Note thatS(H) might depend on a0 measure change ofH. For this reason we introduce a

similar but more complicated notion. Let us denote theσ-algebra

〈σ(S) | S ⊆ [k], k ∈ S, |S| = k − 1〉

by β([k]).

Definition 2.11 Theessentialσ-algebraE(H) is the smallestσ-algebra containingσ([k − 1])∗ in

which all the functions

g(x1, x2, . . . , xk−1) =

∫

xk

f(x1, x2, . . . , xk)t(x1, x2, . . . , xk)

are measurable wheret is an arbitrary bounded function in

L∞(β([k])).

In the next lemmaψ denotes the projection promXk toXk−1 which cancels the last coordinates.

Lemma 2.14 Theσ-algebraE(H) has the following properties

1. E(H) = E(H2) wheneverH△H2 has measure0

2. E(H) is relative separable overσ([k − 1])∗

3. H is almost measurable in〈ψ−1(E(H)), β([k])〉.

4. the functionsfxk:= f(x1, x2, . . . , xk) are measurable inE(H) for almost allxk ∈ X.

10

We will need the following lemma.

Lemma 2.15 LetB1,B2 ⊆ A be two weakly orthogonalσ-algebras on a probability space(X,A, µ)

andW be a subset measurable in〈B1,B2〉. Let furthermoreB3 be theσ-algebra generated by all

the functionsE(ft|B1) wheref is the indicator function ofW and t is in L∞(X,B2). ThenH is

almost measurable in〈B3,B2〉.

Proof. First we prove the statement in the case whenB1 andB2 are independent. LetW2 be a set

with |W2△W | ≤ ǫ of the form

W2 =

n⋃

i=1

(Ai ∩Bi)

where{Bi}ni=1 is a partition ofX into B2 measurable sets andAi ∈ B1 for every1 ≤ i ≤ n. The

existence of such aW2 is guaranteed by the fact thatW is measurable in〈B1,B2〉. We have that

χ(W2) =

n∑

i=1

1

|Bi|χ(Bi)E(χ(W2)χ(Bi)|B1).

whereχ denotes the indicator function of a set. Let

g =

n∑

i=1

1

|Bi|χ(Bi)E(χ(W )χ(Bi)|B1).

Now

‖χ(W2)− g‖1 ≤∥

∥

n∑

i=1

1

|Bi|χ(Bi)E((|χ(W2)− χ(W )|)χ(Bi)|B1)

∥

∥

1.

Therefore by the independence ofB1 andB2 we obtain that‖χ(W2) − g‖1 ≤ ǫ. Together with

‖χ(W2) − f‖1 ≤ ǫ we obtain that‖g − f‖1 ≤ 2ǫ. Sinceg is measurable in〈B3,B2〉 the result

follows.

Now we go to the general case. According to a lemma in... we canconstruct aσ-algebraA ⊆ B2

such thatA is independent fromB1 andW is also measurable in〈B1,A〉. This completes the proof.

Now we are ready to prove lemma 2.14

Proof. The first property is obvious from the definition. From lemma 2.4 we obtain thatE(H) ⊆ B2.

Then lemma 2.8 shows the second condition.

For proving the third statement we use the fact thatσ([k−1]) andβ([k]) are weakly orthogonal.

Then lemma 2.15 implies the statement.

2.8 Relative separable elements and Fourierσ-algebras

Let{Ai}∞i=1 be an infinite sequence of finite Abelian groups. We denote their ultra product[{Ai}∞i=1]

byA. Lemma 2.1 implies thatA is countable compact in theσ-topologyO(A). Lets : A×A → A

denote the addition map defined bys(a, b) = a+b. In general it is not true thats is continuous in the

11

productσ-topologyO(A)×O(A) onA×A but it will be obviously continuous in theσ-topology

O(A×A).

It is clear that theσ-algebraA and the measureµ is invariant under the action ofA

Definition 2.12 LetB be an arbitrary subσ-algebra ofA and letH be an element ofA. We say that

H is relatively separablewith respect toB if there are countable many translatesT = {H+ai | i ∈

N, ai ∈ A} ofH such that theσ-algebra generated byT andB is dense inσ-algebra generated by

all the translates{H+ a | a ∈ A} andB.

The collection of all relative separable elements ofB is again aσ-algebra which containsB. We

denote it byΥ(B). It is clear that ifB is invariant then so isΥ(B). This means that iterating the

operatorΥ starting with the trivialσ-algebra we get an interesting sequence of invariant algebras.

Definition 2.13 (Fourier σ-algebras) Let F0 denote the trivialσ-algebra onA consisting of the

empty set and the whole setA. We define the sequence ofσ-algebrasFi recursively in the way that

Fi = Υ(Fi−1).

Clearly all theσ-algebrasFi are invariant under the action ofA.

2.9 Characterization of the Fourierσ-algebras

In this chapter we give equivalent characterizations of theσ-algebrasFk. Let Dk denote the sub-

group inAk consisting of elements(x1, x2, . . . , xk) with x1 + x2 + . . . + xk = 0. The fac-

tor groupAk/Dk is isomorphic toA where the isomorphism is given byτk(x1, x2, . . . , xk) =

x1 + x2 + . . . + xk. It is also clear thatτk creates a measure preserving equivalence between the

σ-algebrasA(A) andA(Ak,Dk). We prove the following fact which is crucial in this theory.

Lemma 2.16 The mapτk gives a measure preserving equivalence betweenA(Ak,Dk) ∩ σ([k])∗

andFk−1.

The proof will require the next simple lemma.

Lemma 2.17 Let Y be a perfect subgroup in a Abelian groupA. Assume that a measurable set

H ∈ A(A,Y) is relative separable over an invariantσ-algebraB ⊆ A(A). ThenH is also relative

separable overB ∩A(A,Y).

Proof. Let T = {H + ti}∞i=1 be a system of countable many translates ofH such that all other

translates are in theσ-algebra generated byT andB. Let a ∈ A be an arbitrary element andǫ > 0

be a positive number. Then there is a finite expression of the form

Hǫ = ∪ni=1(Ai ∩Bi)

12

with µ(H△Hǫ) ≤ ǫ such thatAi is measurable in theσ-algebra generated byT andBi is in B for

every1 ≤ i ≤ n. Now we have that

g := E(Hǫ|A(A,Y)) =n∑

i=1

χ(Ai)E(Bi|A(A,Y)).

Furthermore‖g−χ(H)‖2 ≤ ǫ sice the distance of two functions is decreased by projections. From

lemma 2.6 it follows that the functionsE(Bi|A(A,Y)) are all measurable inB ∩ A(A,Y). This

completes the proof.

Now we are ready to prove lemma 2.16.

Proof. We prove the statement by induction onk. The casek = 1 is obvious from the definitions.

We assume that it is true fork − 1. Let H be a set inA(A) andf its characteristic function. We

have to prove thatH is measurable inFk−1 if and only if fk is measurable inσ([k − 1])∗. (Recall

thatfk is the characteristic function ofτ−1k (H).)

First assume thatH is measurable inFk−1. This means thatH is relative separable overFk−2.

The sliceσ-algebraS(fk) consists of the translates offk−1 and so it is relative separable over

τ−1k−1(Fk−2. By the induction hypothesisfk−1 is also relative separable overσ([k − 1])∗. This

implies by lemma 2.13 thatfk is measurable inσ([k])∗.

For the other direction we assume thatfk is in σ([k])∗. It is clear thatE(fk) is an invariant

σ algebra on the Abelian groupAk−1. Since almost all the slices offk are measurable inE(fk)

and furthermore all slices are translates offk−1 we get from the invariance ofE(fk) that all the

translates offk−1 are measurable inE(fk). By Lemma 2.14 we obtain thatfk−1 is a relative

separable element overσ([k − 1])∗. It follows from lemma 2.17 thatfk−1 is relative separable over

A(Ak−1,Dk−1)∩σ([k−1])∗ . Applyingτ−1k−1 and the induction hypothesis,H is relative separable

overFk−2.

As a corollary we get the next theorem.

Theorem 1 Let f : X → C a function inL2 and letfk be as in definition 2.6. Then the following

statements are equivalent.

1. f is measurable inFk−1

2. f is orthogonal to everyg withUk(g) = 0.

3. fk is measurable inσ([k])∗

4. fk is orthogonal to everyg withOk(g) = 0.

Lemma 2.18 Theσ-algebrasFi have the projection property for everyi.

2.10 Group extensions

Let G be a second countable compact topological group with the Borel σ-algebraG on it. We

denote the Haar measure onG by ν. A functionf : A → G is called apre-cocycleof orderk if f

13

is measurable inFk andgt(a) := f(a+ t)f(a)−1 is measurable inFk−1 for every fixedt ∈ A.

Two pre-cocyclesf andf2 of orderk are called equivalent iff2 = f3fz wherez is a fixed

element inG andf3 is measurable inFk−1.

Let us define the action ofA onA×G by

a.(b, g) = (b+ a, f(b+ a)f(b)−1g)

and the action ofG onA×G by

h.(b, g) = (b, gh).

The two actions commute with each other. It is clear that the above actions preserves the measure

µ× ν whereµ is the ultra product measure onA. Furthermore theσ-algebraFk−1 ×G is invariant

underA.

Let I denote theσ-algebra ofA invariant sets inFk−1 × G. The action ofG leavesI invariant

and acts on it in an ergodic way. Now Lemma 3.23 in [GLASNER] implies that there is a closed

subgroupH in G such that the action ofG on I is isomorphic to the action ofG on the left coset

space ofH in G. We say thatH is the Mackey group corresponding tof . Note that it is defined up

to conjugacy.

We say thatf is minimal if I is trivial. We introduce the averaging operator

Q : L2(Fk−1 × G) → L2(Fk−1 × G)

by

(Qf)(b, g) =

∫

a

a.(b, g) dµ.

Lemma 2.4 shows thatQ has the required form. It is also clear thatQ is the orthogonal projection

toL2(I).

Lemma 2.19 There is a minimal pre-cocyclef2 : A → H such thatf andf2 are equivalent where

H is the Mackey group off .

Proof. See [GLASNER] 3.25 pg 73

Lemma 2.20 If f : A → G is a minimal pre-cocycle then there is aσ-algebraFk−1 ⊂ H ⊂ Fk

and a measure algebra equivalenceφ : Fk−1 × G → A which commutes with theA action.

Furthermoreφ restricted toFk−1 × Triv(G) gives an equivalence betweenFk−1 andFk−1 ×

Triv(G) whereTriv(G) denotes the trivialσ algebra onG.

Proof. For a Borel setB ⊆ G let T (B) denote the set{g | f(g) ∈ B}. We show thatµ(M ∩

T (B)) = µ(M)ν(B) for any two measurable setsM ∈ Fk−1 andB ∈ G. Sincef is minimal

we have thatQ(H × B2) is (almost everywhere) the constant function with valueµ(M)ν(B2) for

14

any Borel setB2. Lets choose a countable dense setB of open sets inG with respect to the Haar

measure. For almost every pair(a, g) ∈ A×G we have that

∫

b

χM×B2(b.(a, g)) = µ(M)ν(B2)

for everyB2 ∈ B whereχM×B2is the characteristic function ofM × B2. Let (a, g) be one such

pair. then−a.(a.g) = (0, f(−a)f(a)−1g) = (0, g2) is clearly another such pair. The value of

the above integral for(0, g2) is equal toµ(M)ν(B2). On the other hand it is the same asµ(M ∩

T (B2g−12 f(0))). Using the translation invariance ofν this formula implies thatµ(M)ν(B2g

−12 f(0)) =

µ(M ∩ T (B2g−12 f(0))). SinceB is dense we get the formula for everyB ∈ G.

Now it is easy to check thatφ(M×G) =M∩T (B) extends to a measure preserving equivalence

and it is given by

φ(W ) = {a | (a, f(a)) ∈W}.

Definition 2.14 LetFk−1HFk be aA invariantσ-algebra. Then an automorphism of the measure

algebra(H, µ) is called ak- automorphism if σ commutes with the action ofA on itself and leaves

Fk−1 invariant. The set ofk-automorphisms will be denoted byAutk(H). Anyk automorphism

induces an action onL2(H) such thatE(σ(f)|Fk−1) = E(f |Fk−1).

Lemma 2.21 Let f be a minimal pre-cocycle of orderk and letH be theσ-algebra constructed in

lemma 2.20. ThenG induces a faithful action : G→ Autk(H).

Proof. The proof is an immediate corollary of lemma 2.20. The actionof G onA × G induces a

similar action onH.

2.11 Cubic structure

In this section we borrow part of our language from [6]. For a natural numberk let Vk denote the

set of all subsets in[k]. We will think of the elements ofVk as the vertices of ak-dimensional cube.

We will use the notion of ad-dimensionalface in the natural way. Let us define the subgroupBk

in AVk as the collection of vectors(ai)i∈Vk

satisfying all the equations

ap − aq + ar − as = 0

where{p, q} is a1 dimensional face and{p, q, r, s} is a2 dimensional face ofVk. We call the vertex

corresponding to the empty set the0-vertex. Ifv ∈ Vk is a vertex then thespider S(v) is the subset

consisting ofv and its neighbors inVk. The following lemma shows that the projection ofBk to

the coordinates inS(v) gives a bijection betweenBk andAS(v). In particularBk is isomorphic to

Ak+1.

15

Lemma 2.22 Let δk : AS(0) 7→ AVk denote the homomorphism defined by

δk(a0, a1, a2, . . . , ak)v = a0 +∑

i∈v

ai.

Thenδk gives a measure preserving isomorphism betweenAk+1 andBk.

For a vertexv we introduce the projectionτv : Bk → A to the coordinatev. We denote the

σ-algebraτ−1v (A(A)) onBk by Av. ObviouslyAv ⊂ A(Bk). By abusing the notation we also

define the mapsτv on AS(0) by τv(x) = τv(δk(x)). The following lemma shows the connection

between the Gowers norm and the cubic structure.

Lemma 2.23 Letf : A 7→ C be a bounded measurable function. Then

U2k

k (f) =

∫

x∈Bk

∏

v∈Vk

f ǫ(v)(τv(x))

wheref ǫ(v) denotesf whenever|v| is odd and denotesf whenever|v| is even.

Let f = (fv)v∈Vkbe a collection of bounded measurable functions onA. We will need the

2k-form

Uk(f) =

∫

Bk

∏

v∈Vk

fv(τv(x)).

This form naturally extends to the2k-th tensor power ofL∞(A) as a linear function. By abusing

the notation we denote that linear function in the same way.

The next lemma shows the connection of the cubic structure with the Fourierσ-algebras.

Lemma 2.24 For everyv ∈ Vk

Av ∩ 〈{Aw|w 6= v}〉 = τ−1v (Fk−1).

Proof. LetBv denote theσ-algebra〈{Aw|w 6= v}〉 and letB ⊆ A(A) denote the uniqueσ-algebra

with the property thatτ−1v (B) = Bv. The transitivity of the symmetry group of the cube onVk

implies thatB does not depend onv. Let f : A 7→ C be any bounded measurable function. From

lemma 2.23 we get that

U2k

k (f) =

∫

x∈Bk

∏

v∈Vk

E(f ǫ(v)(τv(x))|Bv) = Uk(E(f |B)).

This implies thatFk−1 ⊆ B.

Now we prove thatB ⊆ Fk−1. Using the bijectionδk it suffices to show that iff : A 7→ C is a

bounded measurable function such that onAS(0) the functionτ[k](f) is measurable in theσ-algebra

generated by{τv|v ∈ Vk , v 6= [k]} thenf ∈ Fk−1. This follows easily from theorem 1.

16

2.12 Face actions

Let Fk−1 ⊂ H ⊂ Fk be anA invariantσ-algebra and letσ ∈ Autk(H). Let furthermoreT be a

subset ofVk+1. In this subsection we introduce the actionl(T, σ) on the tensor product⊗VkL∞(H)

defined by

l(T, σ)⊗

v∈Vk+1

fv =⊗

v∈Vk+1

σvfv (1)

where{fv}v∈Vk+1is a collection of boundedH-measurable functions andσv = σ whenever

v ∈ T and is the identity elsewhere. The special case whereT is a face is of special importance.

Such actions will be calledface actions.

Lemma 2.25 If e is an edge ofVk+1 and σ ∈ Autk(A) then actionl(e, σ) preserves the form

Uk+1.

Proof. Without loss of generality we can assume thate = {∅, {k + 1}}. Let {fv}v∈Vk+1be a

collection of bounded real valued functions onA.. Using lemma 2.22we obtain that

Uk+1(l(e, σ)⊗v∈Vk+1fv)) =

=

∫

a0,a1,...,ak+1

σf∅(a0)σf{k+1}(a0 + ak+1)∏

v∈Vk+1\e

fv(a0 +∑

i∈v

ai) =

=

∫

ak+1

∫

a0,a2,...,ak+1

σ(f∅(a0)f{k+1}(a0 + ak+1))∏

v∈Vk+1\e

fv(a0 +∑

i∈v

ai)

The inner integral can be written as

Uk(σf∅ ⊗⊗

v∈Vk\∅

fv) (2)

where

fv(x) = f(x+ ak+1)f(x).

Now lemma 2.24 implies that in (2) the termσf∅ can be replaced by

E(σf∅|Fk−1) = E(f∅|Fk−1)

and then we can replace it byf∅. After this transformation (2) becomes

Uk(⊗

v∈Vk

fv).

This finishes the proof.

17

2.13 Isometric extensions are Abelian

The main result in this section is the following lemma.

Lemma 2.26 (Abelian extension)Letf : A → G be a minimal pre-cocycle of orderk. ThenG is

Abelian.

Proof. Let e1 ande2 be two edges ofVk+1 intersecting each other at a vertexw ∈ Vk+1. Let g1, g2

be two elements fromG. LetH be theσ algebra guaranteed by lemma 2.20. Now we denote bycw

the commutator[l(e1, (g1), l(e2, (g2)]. We have thatcw is acting on⊗Vk+1L∞(H) and the action

is given by

cw = l(w, ([g1, g2])).

From lemma... we ga that the formUk+1 is preserved bycw. This means that the composition

of such actions for a sequence of different verticesw also preservesUk+1. From this we deduce

that([g1, g2]) has to be trivial. Letf be a bounded real valuedH measurable function andg =

f − ([g1, g2])f . Then

Uk+1(g) =

∫

x

∏

v∈Vk+1

(f(τv(x))− ([g1, g2])f(τv(x))).

Expanding the product on the right side all the terms have thesame integral and the number of

positive signs is the same as the number of negative signs. Weget thatUk+1(g) = 0. Sinceg ∈ Fk

this implies thatg = 0.

3 Higher order Fourier analysis

3.1 Definition of higher order Fourier analysis

Thei-th order Fourier analysis deals with the decomposition of the moduleL2(Fk) into irreducible

A invariantL∞(Fk−1)-modules. The fundamental theorem of higher order fourier analysis is the

following.

Theorem 2 (Fundamental Theorem)The rank oneL∞(Fk−1) modules are pairwise orthogonal

and they generate the Hilbert spaceL2(Fk). Every such rank one module has a generator of the

formα : A → C with |α| = 1 andα(x)α(x + t) ∈ Fk−1. These functions are calledk-th order

characters.

It will turn out that the above mentioned rank1 modules are forming an Abelian group using the

point-wise multiplication.

18

3.2 Higher order group algebras

In this section we introduce algebras which replace convolution algebras on ordinary compact topo-

logical groups in the higher order setting. Let firstMk denote the spaceL2(A×A,Fk⊗Fk, µ×µ)

of Hilbert-Schmidt operators. It is easy to check thatMk has a BanachC∗-algebra structure with

multiplication

(K1 ◦K2)(x, y) =

∫

z

K1(x, z)K2(z, y) dµ.

Definition 3.1 (Higher order group algebras) LetCk ⊂ Mk denote the set of elementsK ∈ Mk

such that the functions

fx,y(t) := K(x+ t, y + t)

are inL2(Fk−1) for everyx, y ∈ A.

Lemma 3.1 The setCk is forming a subC∗-algebra ofMk.

Proof. Use lemma 2.4.

Now we describe a useful way of constructing elements inDk.

Lemma 3.2 Let f, g be two functions inL2(Fk). Then there is an elemente = e(f, g) in Ck such

that for every fixedx, y

e(x+ t, y + t) = E(f(x+ t)g(y + t)|Fk−1)(t).

3.3 Decomposition of the elements inCk

Let C be a self adjoint element ofCk ∩ L∞(A × A). We denote byim(C) ⊂ L2(A) the image

space of the operatorC. Note thatim(C) is a separable Hilbert subspace of the non separable space

L2(A). It is classical that

im(C) =⊕

Vi

whereVi are finite dimensional eigenspaces ofC corresponding to different eigenvectors. The

spacesVi are contained inL∞(A). The main result of this section is the following.

Lemma 3.3 Each spaceVi is contained in a finite rank module overL∞(Fk−1). Furthermore this

finite rank module can be decomposed into rank1 modules.

Proof. Let Pi denote the orthogonal projection toVi. The operatorsPi are contained in the

Banach algebra generated byC so they are all elements ofCk. Let f1, f2, . . . , fd ∈ L∞(A) be an

orthonormal basis forVi. We also introduce the vector valued functionf : A 7→ Cd defined by

f(x) = (f1(x), f2(x), . . . , fd(x)). The operator kernelPi satisfiesPi(x, y) = 〈f(x), f(y)〉. Since

Pi ∈ Ck we get that〈f(x), f(x + t)〉 ∈ Fk−1 for every fixedt ∈ A.

19

We show that there are elementst1, t2, . . . , td such that the functionrk(f(x + t1), f(x +

t2), . . . , f(x+ td)) is equal tod on a positive measure set inFk. Let

h(t1, t2, . . . , td) := |det(f(t1), f(t2), . . . , f(td))|2.

It is easy to see that the linear independence off1, f2, . . . , fd implies thath > 0 on a positive

measure subset set inAd. This implies that there is a fixed vectort1, t2, . . . , td such thath(t1 +

x, t2 + x, . . . , td + x) > 0 on a positive measure set ofA. Since the matrixWi,j(x) = 〈f(ti +

x), t(tj , x)〉 is measurable inFk−1 we get that the set where

h(t1 + x, t2 + x, . . . , td + x) = det(Wi,j(x))

is positive is onP ∈ Fk−1.

The next step is that we run the Gram-Schmidt orthogonalization on the setP for the vectors

{f(x + ti)}di=1. It is easy to see that all the coefficients are measurable inFk−1 so we find func-

tionso1, o2, . . . , od : P → Cd such that{oi(x)}di=1 is an orthonormal basis for everyx ∈ P and

furthermore

oi =

d∑

j=1

λi,j(x)f(x + tj)

for every1 ≤ i ≤ d whereλi,j(x) is measurable in{k−1. From this we get that every translate

f(x+ t) of f can be expressed onP from {oi}di=1 usingFk−1 measurable functions as coefficients.

Now we use Rohlin’s lemma to find at most countable many subsetsP1, P2, . . . of P all mea-

surable inFk−1 such thatA is a disjoint union of translated versions of these sets. By translating

the basis{oi}∞i=1 correspondingly one obtains an extension of the orthonormal basis{oi}∞i=1 to the

whole setA. In this extended basis any translate off can be expressed withFk−1 measurable

coefficients. This shows thatVi is contained in a module of rank at mostd.

Let O(x) denote the unitary matrix formed by the{oi(x)}di=1. It is clear thatx → O(x) is a

pre cocycle of degreek. This means by lemma 2.19 and lemma 2.26 that it is equivalentwith a

cocycleO′(x) going to some abelian subgroupH of the unitary group. By decomposingH into 1

dimensional irreducible representations we find pre-cocyclesO′i(x) taking values inC of absolute

value1. This provides rank one modules overFk−1 that generateVi.

Corollary 3.1 LetC ∈ Ck ∩ L∞(A×A) be an arbitrary element. Thenim(C) is contained in a

space generated by rank1 modules.

Proof. Using thatim(C∗C) = im(C) we can assume thatC is self adjoint. Then the previous

lemma completes the proof.

20

3.4 Proof of the fundamental theorem

Let Fk−1 ⊂ H ⊂ Fk be anA invariantσ algebra that is relative separable with respect toFk−1.

Let furthermoref1, f2, . . . be a relative orthonormal basis ofL2(H) overFk−1. This means that

E(fifj |Fk−1) is the constant1 function for i = j and is the constant0 function for i 6= j. Every

elementf ∈ L2(H) can be uniquely written as∑

i λi(x)fi where the functionsλi areFk−1 mea-

surable and can be obtained byλi = E(ffi|Fk−1). Now let us introduce the uniquely determined

functionsλi,j(t, x) by

fi(x+ t) =∑

j

λi,j(t, x)fj(x). (3)

Let furthermoreMx,y denote theN× N matrix defined by

Mx,y(i, j) = λi,j(y − x, x).

Lemma 2.18 guarantees thatMx,y is a measurable function of(x, y). It is easy to check from (3)

that

Mx,yMy,z =Mx,z

hold for almost all triplesx, y, z

Mx,y =M−1y,x =M∗

y,x

holds for almost all pairsx, y. This implies that there is a fixedy such that the above equations hold

for almost allx andz. LetMx := Mx,y. We have thatMx,z = MxM−1z for almost allx, z. By

definition we have thatMxM−1x+t is measurable inFk−1 for every fixedt. This guarantees thatMk

is measurable inFk since theσ-algebra generated byMx has to be relative separable with respect

toFk−1. LetH′ be theσ-algebra generated by the functionx→Mx. It is easy to see thatH ⊆ H′.

Summarizing the above information we obtain thatMx,y(i, j) ∈ Ck for every fixedi, j. Using

Corollary 3.1 we get thatMx,y is measurable in aσ algebra generated by rank1 modules. So the

σ-algebraH′ is generated by rank1 modules. SinceFk is the union of all relativeFk−1 separable

invariantσ-algebras the proof is complete.

Balazs Szegedy University of Toronto, Department of Mathematics, St George St. 40, Toronto, ON,

M5R 2E4, Canada

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