Proof of the structure theorem

We can now complete the proof of the structure theorem by using the arguments of [18,

§7 and§8] more or less verbatim. In fact these arguments can be abstracted as follows.

Theorem7.1. (Abstract structure theorem) Let I be an interval bounded by O(1).

Let ν:X!R⁺ be any measure, and let f7!Dfe be a (non-linear) operator obeying the following properties:

• if the pointwise bound|f|6ν+1 holds, thenDfe :X!I takes values in I, in par-ticular

Dfe =O(1); (53)

• if 16K61/η6 and f1, ..., fK:X!Rare functions satisfying the pointwise bound

|fk|6ν+1,k∈[K],then (49)holds.

Then, for any g:X!R⁺ with the pointwise bound 06g6ν, there exist functions g_U^⊥, gU:X!Robeying estimates (26), (27) and (28),such that

Z

X

gUDge U

6η4. (54) Indeed, Theorem 4.7 immediately follows by applying Theorem 7.1, with (29) fol-lowing from (54) and (48).

In the remainder of this section we prove Theorem 7.1. Henceforth we fixI, ν and De obeying the hypotheses of the theorem.

7.2. Factors

As in [18], we shall recall the very useful notion of factor from ergodic theory, though for our applications we actually only need the finitary version of this concept.

Let us set X to be the probability spaceX=(X,BX, µ_X), whereX=Z_N, BX=2^X is the power set of X and µ_X is the uniform probability measure on X. We define a factor(³¹) to be a quadrupleY=(Y,BY, µ_Y, π_Y), where (Y,BY, µ_Y) is a probability space (and thusBY is aσ-algebra onY andµ_Y is a probability measure onBY) together with a measurable map π:X!Y such that (π_Y)_∗µ_X=µ_Y, or in other words µ_X(π⁻¹_Y (E))=

µ_Y(E) for all E∈B_Y. The factor mapπ_Y induces the pullback mapπ_Y^∗:L²(Y)!L²(X) and its adjoint (π_Y)_∗:L²(X)!L²(Y), whereL²(X) is the usual Lebesgue space of square-integrable functions on X. We refer to the projectionπ_Y^∗(π_Y)_∗:L²(X)!L²(X) as the conditional expectation operator, and denote π^∗_Y(π_Y)_∗(f) by E(f|Y); this is a linear self-adjoint orthogonal projection fromL²(X) toπ_Y^∗L²(Y).

(³¹) In infinitary ergodic theory one also requires the probability spacesXandYto be invariant under the shiftT, and for the factor mapπto respect the shift. In the finitary setting it is unrealistic to demand these shift-invariances, for ifNwere prime then this would mean that there were no non-trivial factors whatsoever. While there are concepts of “approximate shift-invariance” which can be used as a substitute, see [33], we will fortunately not need to use them here, as the remainder of the argument does not even involve the shiftT at all.

The conditional expectation operator is in fact completely determined by the σ-algebra π⁻¹_Y (BY)⊂BX. Since X is finite (with every point having positive measure), π_Y⁻¹(BY) is generated by a partition of X into atoms (which by abuse of notation we refer to as atoms of the factorY), and the conditional expectation is given explicitly by the formula

E(f|Y)(x) =E_y∈B(x)f(y),

where B(x) is the unique atom ofπ⁻¹(B_Y) which contains x. We refer to the number of atoms ofYas the complexity of the factorY.(³²) By abuse of notation, we say that a functionf:X!R ismeasurable with respect to Y if it is measurable with respect to π_Y⁻¹(BY), or equivalently if it is constant on all atoms ofY. Thus for instance (πY)^∗L^q(Y) consists of the functions inL^q(X) which are measurable with respect toY.

IfY=(Y,B_Y, µ_Y, π_Y) andY⁰=(Y⁰,B_Y⁰, µ_Y⁰, π_Y⁰) are two factors, we may form their joinY∨Y⁰=(Y×Y⁰,B_Y×B_Y⁰, µ_Y×µ_Y⁰, π_Y⊕π_Y⁰) in the obvious manner; note that the atoms of Y∨Y⁰ are simply the non-empty intersections of atoms of Y with atoms of Y⁰, and so any function which is measurable with respect to Y or Y⁰ is automatically measurable with respect toY∨Y⁰.

Note that any functionf:X!Rautomatically generates a factor (R,B_R, f_∗µ_X, f), whereB_Ris the Borel σ-algebra, which is the minimal factor with respect to whichf is (Borel-)measurable. In our finitary setting it turns out that we need a discretized version of this construction, which we give as follows.

Proposition 7.3. (Each function generates a factor) For any function G:X!I there exists a factor Y(G)with the following properties:

• (Glies in its own factor) for any factor Y⁰,

G=E(G|Y(G)∨Y⁰)+O(η₄²); (55)

• (bounded complexity)Y(G)has at most Oη4(1) atoms;

• (approximation by continuous functions of G) if A is any atom in Y(G), then there exists a polynomial Ψ_A:R!Rof degree O_η5(1)with coefficients O_η5(1) such that

ΨA(x)∈[0,1] for all x∈I (56)

and Z

X

|1A−ΨA(G)|(ν+1)η5. (57)

(³²) It would be more natural to work instead with theentropy ofYrather than the complexity, but the entropy is a slightly more technical concept and so we have avoided its use here for simplicity.

Proof. This is essentially [18, Proposition 7.2], but we shall give a complete proof here for the convenience of the reader.

We use the probabilistic method. Letαbe a real number in the interval [0,1], chosen at random. We then define the factor

Y(G) := (R,B_η2

4,α, G_∗µX, G), whereB_η2

4,α is theσ-algebra on the real line Rgenerated by the intervals [(n+α)η₄²,(n+α+1)η²₄)

forn∈Z. This is clearly a factor ofX, with atoms

An,α:=G⁻¹([(n+α)η²₄,(n+α+1)η₄²)).

Since G ranges in I, and we allow constants to depend on I, it is clear that there are at mostOη₄(1) non-empty atoms and thatGfluctuates by at mostO(η₄²) on each atom, which yields the first two desired properties. It remains to verify that with positive prob-ability, the approximation by continuous functions property holds for all atomsAn,α. By the union bound, it suffices to show that each individual atomAn,αhas the approxima-tion property with probability 1−O(η5).

By the Weierstrass approximation theorem, we can for each α find a polynomial ΨA_n,α obeying (56) which is equal to 1_[(n+α)η2

4,(n+α+1)η₄²)+O(δ) outside of the set En,α:= [(n+α−η₅²)η₄²,(n+α+η₅²)η²₄]∪[(n+α+1−η₅²)η₄²,(n+α+1+η²₅)η₄²].

Simple compactness arguments allow us to take ΨA_n,α to have degreeOη₅(1) and coeffi-cientsOη₅(1). Since

1A_n,α= 1_[(n+α)η2

4,(n+α+1)η²₄)(G), we thus conclude (from (15)) that

Z

X

|1A−ΨAn,α(G)|(ν+1)η₅+ Z

X

1_En,α(G)(ν+1).

By Markov’s inequality, it thus suffices to show that Z 1

0

Z

X

1E_n,α(G)(ν+1)

dαη₅².

But this follows from Fubini’s theorem, (15) and the elementary pointwise estimate Z 1

0

1E_n,α(G)dαη₅².

Henceforth we set Y(G) to be the factor given by the above proposition. A key consequence of the hypotheses of Theorem 7.1 is thatν−1 is well distributed with respect to any finite combination of these factors.

Proposition 7.4. (ν uniformly distributed with respect to dual function factors) Let K>1 be an integer with K=Oη₄(1), and let f1, ..., fK:X!R be functions with the pointwise bounds |fk|6ν+1for all k∈[K]. Let Y:=Y(Dfe 1)∨...∨Y(Dfe K). Then

Dfe _k=E(Dfe _k|Y)+O(η²₄) (58) for all k∈[K], there is a Y-measurable set Ω⊂X obeying the smallness bound

Z

X

1_Ω(ν+1)_η4η₅^1/2 (59)

and we have the pointwise bound

|(1−1Ω)E(ν−1|Y)|6Oη₄(η^1/2₅ ). (60) Proof. We repeat the arguments from [18, Proposition 7.3]. The claim (58) follows immediately from (55), so we turn to the other two properties. Since each Y(Dfe _k) is generated byO_η4(1) atoms,Yis generated byO_η4,K(1)=O_η4(1) atoms. Call an atomA ofY small ifR

X1A(ν+1)6η^1/2₅ , and let Ω be the union of all the small atoms, then Ω is clearlyY-measurable and obeys (59). It remains to prove (60), or equivalently that

R

X1A(ν−1) R

X1A

=Ey∈Aν(y)−1η₄η^1/2₅ +o(1) for all non-small atomsA.

Fix a non-small atomA. SinceA is not small, we have Z

X

1_A(ν−1)+2 Z

X

1_A= Z

X

1_A(ν+1)> η^1/2₅ . Hence it will suffice to show that

Z

X

1_A(ν−1)_η4η₅+o(1).

On the other hand, asAis the intersection of atomsA₁, ..., A_KfromY(Dfe ₁), ...,Y(Dfe _K), we see from Proposition 7.3 and an easy induction argument that there exists a polyno-mial Ψ:R^K!Rof degreeO_η5(1) with coefficientsO_η5(1) which mapsI^K into [0,1] such

that Z

X

|1A−Ψ(Dfe 1, ...,Dfe K)|(ν+1)η₄η5.

In particular,

Z

X

(1A−Ψ(Dfe 1, ...,Dfe K))(ν−1)η4η5.

On the other hand, by decomposing Ψ into monomials and using (49) (assuming η₆ sufficiently small depending onη₅), we have

Z

X

Ψ(Dfe 1, ...,Dfe K)(ν−1) =o(1)

and the claim follows (we can absorb theo(1) error by takingN large enough).

7.5. The inductive step

The proof of the abstract structure theorem proceeds by a stopping time argument. To clarify this argument we introduce a somewhat artificial definition.

Definition 7.6. (Structured factor) Astructured factor is a tuple YK= (YK, K, F1, ..., FK,ΩK),

where K>0 is an integer, F1, ..., FK:X!R are functions with the pointwise bounds

|Fk|6ν+1 for allk∈[K],YK is the factorYK:=YK(F1)∨...∨YK(FK) and ΩK⊂X is a YK-measurable set. We refer toKas theorder of the structured factor, and ΩK as the exceptional set. We say that the structured factor hasnoise level σ for someσ >0 if we have the smallness bound

Z

X

1Ω_K(ν+1)6σ and the pointwise bound

|(1−1Ω_K)E(ν−1|YK)|6σ. (61) Ifg:X!Ris the function in Theorem 7.1, we define theenergyE_g(Y_K) of the structured factorY relative tog to be the quantity

Eg(YK) :=

Z

X

(1−1Ω_K)E(g|YK)².

IfYK has noise level σ61, then, sinceg is bounded in magnitude byν,

|(1−1Ω_K)E(g|YK)|6(1−1Ω_K)(E(ν−1|YK)+1)61+σ62, (62) and so we conclude the energy bound

06Eg(YK)64. (63)

This will allow us to apply anenergy increment argument to obtain Theorem 7.1. More precisely, Theorem 7.1 is obtained from the following inductive step.

Proposition 7.7. (Inductive step) Let YK=(YK, K, F1, ..., FK,ΩK) be a struc-tured factor of order K with noise level 0<σ <η⁴₄. If we set

FK+1:= 1

1+σ(1−1Ω_K)(g−E(g|Y)) (64)

and we suppose that

Z

X

FK+1DFe K+1

> η4, (65)

then there exists a structured factor YK+1=(YK+1, K+1, F1, ..., FK, FK+1,ΩK+1) of order K+1 with noise level σ+Oη₄(η^1/2₅ )satisfying the energy increment property

Eg(YK+1)>Eg(YK)+cη²₄ (66) for some constant c>0 (depending only on I).

Let us assume Proposition 7.7 for the moment and deduce Theorem 7.1. Starting with a trivial structured factorY0 of order 0, and iterating Proposition 7.7 repeatedly (and using (63) to prevent the iteration for proceeding for more than 4/cη₄²=Oη₄(1) steps), we may find a structured factorYK of orderK=Oη₄(1) with noise level

σ=Oη₄(η₅^1/2)< η₄⁴, (67) such that the functionFK+1defined in (64) obeys the bound

Z

X

FK+1DFe K+1

6η4. If we thus setg_U:=F_K+1 and

g_U⊥:= 1

1+σ(1−1_ΩK)E(g|Y),

then we easily verify (26) and (54), while (27) follows from (61), since E(g|Y)61+E(ν−1|Y).

To prove (28), we see from (67) that it suffices to show that Z

X

(1−1Ω_K)E(g|Y) = Z

X

g−Oη₄(η₅^1/2).

Since Ω_K is Y-measurable, the left-hand side is R

Xg−R

X1_ΩKg. But the claim then follows from (61) and (67). This proves Theorem 7.1.

It remains to prove Proposition 7.7. Set

YK+1:=Y∨Y(DFe K+1) =Y(DFe 1)∨...∨Y(DFe K+1).

Now, by Proposition 7.4, we can find a Y_K+1-measurable set Ω obeying the smallness bound (59) and the pointwise bound

|(1−1Ω)E(ν−1|YK+1)|6Oη₄(η₅^1/2). (68) Set ΩK+1:=ΩK∪Ω. This is still YK+1-measurable and R

XΩK+16σ+Oη₄(η^1/2₅ ); from (68), we thus conclude thatYK+1 has noise levelσ+Oη4(η₅^1/2). Thus the only thing left to verify is the energy increment property (66).

From (64) and (65) we have

By (53) and the Cauchy–Schwarz inequality, we conclude that Z

for somec>0.

To pass from this to (66), first observe from (62) and (59) that Z

X

(1ΩK+1−1ΩK)E(g|YK)²η4η₅^1/2,

and so, by the triangle inequality and (63), (66) will follow from the estimate Z

X

(1−1Ω_K+1)E(g|YK+1)²>

Z

X

(1−1Ω_K+1)E(g|YK)²+2cη²₄−O(η₄³).

Using the identity

E(g|Y_K+1)²=E(g|Y_K)²+

E(g|Y_K+1)−E(g|Y_K)

2+2E(g|Y_K)(E(g|Y_K+1)−E(g|Y_K)) and (70), it will suffice to show that

Z

X

(1−1Ω_K+1)E(g|YK)(E(g|YK+1)−E(g|YK))η₄³.

Now observe thatE(g|YK+1)−E(g|YK) is orthogonal to allYK-measurable functions, and in particular

Z

X

(1−1_ΩK)E(g|Y_K)(E(g|Y_K+1)−E(g|Y_K)) = 0.

Thus, it suffices to show that Z

X

(1Ω_K+1−1Ω_K)E(g|YK)(E(g|YK+1)−E(g|YK))η³₄.

Since everything here isY_K+1-measurable, we may replaceE(g|Y_K+1) byg. Using (62), it then suffices to show that

Z

X

(1_ΩK+1−1_ΩK)

g−E(g|Y_K) η₄³.

But this follows from the pointwise bound 06g6ν, from (62) and (59). This concludes the proof of Proposition 7.7, which in turn implies Theorem 7.1 and thus Theorem 4.7.

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