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Gowers Norms, Host-Kra Factors and Nilmanifolds

2.5 The Ergodic Analogue

2.5.2 Gowers Norms, Host-Kra Factors and Nilmanifolds

Recall the definition of higher-degree uniformity norms in arithmetic combinatorics, which originated in Gowers’s work on Szemer´edi’s Theorem for longer progressions [Gow01].

Definition 2.2. Let G be a finite Abelian group. For any positive integer k ≥2 and any function f :G→C, define the Uk-norm by the formula

kfk2Ukk :=Ex,h1,...,hk∈G

Y

ω∈{0,1}k

C|ω|f(x+X

i

ωihi),

where C|ω|f =f if P

iωi is even and f otherwise.

By a special case of Proposition 2.4, which was in fact proved in [Gow01], the Uk+1 -norm governs the average over arithmetic progressions of length k (this is because progressions of length k have Cauchy-Schwarz complexity k−2). A family of semi-norms analogous to the Uk-norms have recently appeared in the work of Host and Kra [HK05].

Definition 2.48. For f ∈L(µ) and k∈N, we define the Host-Kra semi-norms as

|||f|||k :=

Z

X[k]

f⊗ · · · ⊗f dµ[k]

1/k

.

Of course we haven’t actually defined the measure µ[k] yet, nor the space X[k] over which we integrate. The definition below looks rather off-putting, and we invite the reader to skip the details on first reading. However, even on more superficial inspection it can be intuited that the construction of the measure µk encodes the structure of combinatorial cubes of dimension k.

2.5 The Ergodic Analogue

Definition 2.49. Let X[k] =X2k and define T[k]: X[k]→X[k] by T[k]=T × · · · ×T (2k times). We write a point x ∈ X[k] as x = (x)∈{0,1}k and make the natural identification of X[k+1] with X[k]×X[k], writing x = (x0,x00) for a point of X[k+1], with x0,x00 ∈ X[k]. By induction, we define a measure µ[k] on X[k] invariant under T[k]. Set µ[0] :=µ. Let I[k] be the invariant σ-algebra of (X[k],X[k], µ[k], T[k]). Then µ[k+1] is defined to be the relatively independent joining of µ[k] with itself over I[k], meaning that if F and G are bounded functions on X[k],

Z

X[k+1]

F(x0)·G(x00)dµ[k+1](x) = Z

X[k]

E(F|I[k])(y)·E(G|I[k])(y)dµ[k](y) . Since (X,X, µ, T) is assumed to be ergodic, I[0] is trivial and µ[1] =µ×µ. Just like the Uk-norms in arithmetic combinatorics, these seminorms are nested, in the sense that they satisfy

|||f|||1 ≤ |||f|||2 ≤ · · · ≤ |||f|||k ≤ · · · ≤ kfk,

and a Gowers-Cauchy-Schwarz-type inequality holds, that is,

Y

∈{0,1}k

f(x)dµ[k]

≤ Y

∈{0,1}k

|||f|||k,

which can be used to show that |||.|||k is indeed a semi-norm on L(µ). Moreover, it can be checked that just like the Uk-norms, the semi-norms |||.|||k can be defined inductively via the formula

|||f|||2k+1k+1 = Z

Ik

E(f⊗2k|I[k])2[k].

Together with the Von Neumann Ergodic Theorem, which states that for an er-godic system (X,X, µ, T) and f ∈ L2(µ), the L2-limit as N tends to infinity of

1 N

PN

n=1f(Tnx) is the constant function R

f dµ, this can be rewritten as

|||f|||2k+1k+1 = lim

N→∞

1 N

N

X

n=1

|||f ·Tnf|||2kk.

This fact in turn is a useful ingredient in the proof of Proposition 2.50 below, which represents the analogue of Theorem 2.4 and will be discussed in more detail at the start of Section 2.5.3. We refer the keen reader to page 20 of [Kra06] for a proof in the case of arithmetic progressions.

Proposition 2.50. Assume that (X,X, µ, T)is ergodic and let d, k, m∈N. Suppose

2.5 The Ergodic Analogue

kfik ≤ 1 for all i = 1,2, . . . , m, and that the system L = (L1, L2, . . . , Lm) in d variables has Cauchy-Schwarz complexity k. Then

lim sup

N→∞

1 Nd

N−1

X

n1,n2,...,nd=0

TL1(n1,...,nd)f1(x) . . . TLm(n1,...,nd)fm(x) 2

min

l=1,2,...,m|||fl|||k+1. With the definitions in place, it is now straightforward to define the sequence of so-called Host-Kra factors, which first appeared in [HK05].

Definition 2.51. Given a measure-preserving system (X,X, µ, T), there is a nested sequence of factors Zk of X such that for any bounded function f on X

|||f|||k+1 = 0 if and only if E(f|Zk) = 0.

It follows straight from this definition combined with Proposition 2.50 that the factors Zkare characteristic for systems of Cauchy-Schwarz complexityk. In particular,Z1 is characteristic for the average along 3-term progressions, while the factor Z2 controls 4-term progressions.

Let us pause for a moment to compare this situation with our combinatorial approach:

In order to concentrate on the structured part in arithmetic combinatorics, we needed a deepU3-inverse theorem which allowed us to decompose any bounded function into a quadratically structured and a quadratically uniform part. In ergodic theory, the fact that the factors Zk are characteristic for systems of Cauchy-Schwarz complexity k follows straight from the definition and Proposition 2.50. The real difficulty lies in giving a geometric description of the factors defined in this very “soft” way.

Having said that, it is not hard to see that the first factor in this sequence Z1

corresponds to the classical Kronecker factor. There are many equivalent ways of describing the Kronecker factor K of a measure-preserving system which do not use the semi-norm |||.|||2.

• K is the largest abelian group rotation factor.

• K is the smallest sub-σ-algebra of X with the property that every member of I[1] is measurable with respect to K ⊗ K.

• The measure µ[2] is relatively independent with respect to K4 and the factor K of X is minimal with this property.

Example 2.52. Let X = T ×T be equipped with the Borel σ-algebra and Haar

2.5 The Ergodic Analogue

measure. Fix α ∈T and define T: X →X by

T(x, y) = (x+α, y+x)

The system is ergodic if and only if α /∈ Q, and it is not isomorphic to a group rotation. The Kronecker factor of X is the factor T equipped with the rotation x 7→

x+α. We say X is a skew extension of T by another copy of T.

It is not hard to see directly that|||f|||2 equals thel4-norm of the Fourier transform of f projected onto the Kronecker factor, and that the Kronecker factor is characteristic for studying ergodic averages along 3-term progressions (see page 21 of [Kra06]). This corresponds to saying that ordinary Fourier analysis suffices in this case.

In order to study longer progressions, higher-order factors are needed. The Conze-Lesigne factor, which in modern terminology represents the second level in the series of Host-Kra factors, was introduced by Conze and Lesigne in a series of papers [CL84], [CL87], [CL88]. Equivalent and more explicit descriptions were given by Rudolph [Rud95] and Host and Kra [HK01], and we refer the interested reader to these works for more detail.

It turns out that every Conze-Lesigne system is the inverse limit of a sequence of 2-step nilsystems (see Theorem 18 in [HK04]). More generally, Host and Kra proved the following deep structure theorem in [HK05]:

Theorem 2.53. For each integer k, the factor Zk is isomorphic to an inverse limit of k-step nilsystems.

In order to make use of this structure theorem, we need to understand what ak-step nilsystem is, as well as what it means to be an inverse limit of a sequence of such systems.

Definition 2.54. Let G be a group. If g, h ∈ G, let [g, h] = g−1h−1gh denote the commutator of g and h. If A, B ⊂G, we write [A, B] for the subgroup of G spanned by {[a, b] :a∈A, b∈B}. The lower central series

G=G1 ⊃G2 ⊃ · · · ⊃Gj ⊃Gj+1 ⊃. . .

of G is defined by setting G1 =G and Gj+1= [G, Gj] for j ≥1. We say that G is k-step nilpotent ifGk+1 ={1G}. IfGis ak-step nilpotent Lie group andΓis a discrete co-compact subgroup, the compact manifold X = G/Γ is a k-step nilmanifold. The group G acts naturally on X by left translation, that is if a∈G and x∈X, then the

2.5 The Ergodic Analogue

translation Ta by a is given by Ta(xΓ) = (ax)Γ. There is a unique Borel probability measure µ (the Haar measure) on X that is invariant under this action. For a fixed element a∈G, we say that the system (G/Γ,G/Γ, Ta, µ) is a k-step nilsystem.

Important examples of nilsystems include the circle nilflow (Example 2.43, easily seen to be a 1-step nilsystem by setting G = R and Γ = Z in the above definition), the skew torus (Example 2.52, a primitive 2-step nilsystem), and the Heisenberg nilflow, which we shall discuss in Example 2.55 below. More information on these basic examples can be found in both [Kra06] and [GT06c].

Example 2.55. Let G be the Heisenberg group R×R×R with multiplication given by

(x, y, z)∗(u, v, w) = (x+u, y +v, z+w+xv),

which is a 2-step nilpotent Lie group (and is perhaps more easily thought of as the group of upper-diagonal real matrices with 1s on the diagonal). Take the discrete co-compact subgroup Γ =Z×Z×Z, so that X =G/Γ is a 2-step nilmanifold. Then the transformation T defined as translation by (g1, g2, g3)∈G together with the Borel σ-algebra X and Haar measure µdefines a 2-step nilsystem. This system is ergodic if and only if g1 and g2 are rationally independent. The compact abelian group G/G2Γ is isomorphic to T2, and the rotation by (g1, g2) on T2 is ergodic. This factor of X represents the Kronecker factor Z1.

It is not terribly important to us to know what exactly an inverse limit is, since it behaves well enough to always allow us to concentrate on a single nilmanifold, but for the sake of completeness we present the definition below.

Definition 2.56. The system (X,X, µ, T) is an inverse limit of a sequence of fac-tors {(Xj,Xj, µj, T)}j∈N if {Xj}j∈N is an increasing sequence of T-invariant sub-σ-algebras such that W

j∈NXj =X up to sets of measure. If each system (Xj,Xj, µj, T) is isomorphic to a k-step nilsystem, then (X,X, µ, T) is an inverse limit of k-step nilsystems.

As indicated earlier, nilmanifolds possess an enormous amount of structure, so by reducing to the study of averages on nilmanifolds via Proposition 2.50 and Theorem 2.53, many questions about the convergence of ergodic averages on abstract measure-preserving systems become explicit computations. Before we look at the general case, however, let us consider in more detail the simple 2-step nilsystem that is the skew torus.

2.5 The Ergodic Analogue