Unit 4, continued: Preserving the required independence of three auxiliary groups

This part studies Definition 5.1, Lemma 5.2, Proposition 5.3, and Corollary 5.4 in B §5, pp.14–18. The preceding parts established the selected law, its conditional parameter laws, and a finite tensor-projection estimate. We now use them to ask which independence relations survive when the averaging length depends on the starting point.

We additionally assume mixing of every finite order, as stated in B (2.1). The paper imports this assertion from R, Rokhlin's multiple-mixing problem for one transformation. A step-by-step reconstruction of its main proof is given in the companion reading on multiple mixing. It remains an input to this article; the definition of ordinary mixing does not supply it.

1. The precise structure: independent auxiliary groups, with the original variable independent of any two

1.1 Auxiliary factors with specified speeds

Retain the unit-roof suspension $C=X\times[0,1)$. Let $Y\to C$ be a probability-preserving flow extension, with original factor $\pi_Y$. There are also finitely many measure-preserving maps

where $q_\alpha>0$ is rational and equivariance is understood modulo null sets at each fixed time. Divide the auxiliary factors into three groups $\mathcal D_1,\mathcal D_2,\mathcal D_3$, allowing empty groups.

Let $x_0$ be the spatial coordinate of $\pi_Y$ and $x_\alpha$ that of $D_\alpha$, with phases $u_0,u_\alpha$. Write $V$ for the full phase array.

1.2 The two conditional laws in Definition 5.1

The paper calls such a system a channel. We work directly with its two defining conditions.

Conditional on $V$, all auxiliary spatial variables have a product law:

For every proper subset $J\subsetneq\{1,2,3\}$, the original variable together with these groups also has a product law conditional on $V$:

The right sides do not depend on the phase values. The case $J=\varnothing$ requires $x_0$ to be independent of all phases. The definition does not require $x_0$ to be independent of all three auxiliary groups jointly.

The suspension $C$ without auxiliary variables satisfies these conditions and provides a starting system.

2. The same product laws hold given the maximal distal factor

Source: B Lemma 5.2, pp.14–15.

2.1 Fixing the phases at an integer time

The phase factor $V$ is a finite-dimensional torus translation with its actual joint phase distribution. It is therefore compact and contained in $W_Y$. The phase law need not be Haar product measure. Characters remain eigenfunctions under translation, and their finite linear combinations are dense in $L^2$ of that actual distribution, giving the required compact structure.

Choose a positive integer $h$ such that every $hq_\alpha$ is an integer. At time $h$ the phases are fixed, the original spatial variable evolves by $T^h$, and the auxiliary variables evolve by $T^{hq_\alpha}$.

Each spatial tuple in (G1) or (G2) has a fixed product measure conditional on the phases. Its transformation is a product of finitely many positive integer powers $T^{e_j}$.

2.2 Ordinary mixing suffices here

If $T$ is mixing, every nonzero integer power is mixing. A finite product of mixing systems is mixing for the product measure: first factor the correlation of two tensor functions coordinatewise, then approximate bounded functions to obtain the general case.

Write the spatial tuple's fixed product measure as $\kappa$ and its time-$h$ product transformation as $R$. The joint phase–spatial law is $\nu_V\otimes\kappa$, with action $(v,z)\mapsto(v,Rz)$. If bounded functions $F,G$ satisfy

then, for each such $v$, mixing of the product system implies

The correlation is bounded in absolute value by $\|F\|_\infty\|G\|_\infty$. Dominated convergence over $v$ makes its $L^2(\nu_V)$ norm tend to zero. This implies the Cesàro relative weak-mixing condition, without requiring a mixing rate uniform in the phases.

Thus the spatial tuple together with the phase factor is relatively weakly mixing over the fixed-phase system. The factor $W_Y$ remains distal at time $h$ and is relatively distal over $V$. Relative disjointness, as stated in Part II, forces the spatial tuple and $W_Y$ to be independent conditional on $V$.

Thus (G1) and (G2) remain valid when the conditioning changes from $V$ to $W_Y$. This is an application of relative disjointness, not a general rule that extra conditioning preserves independence.

For a further flow extension $Y'\to Y$, pulling back the original variables, auxiliary variables, and phases preserves their joint law, so (G1) and (G2) still hold. Repeat the argument for the potentially larger $W_{Y'}$ to obtain the same conclusion on the extension.

3. The assertion of Proposition 5.3

Fix $b\in I=\{1,2,3,4\}$. On

take $\pi_Y(y_b)$ as the original factor. The new group $j$ consists of

with new speeds $iq_\alpha/b$, still positive and rational.

The one-coordinate marginals already make these factors measure-preserving, and their different-speed equivariance follows from the definition. The remaining task is to prove that the new variables satisfy (G1) and (G2) conditional on the new phases.

3.1 Reduction to centered spatial functions

Test the required laws with bounded spatial products and continuous functions of the new phases. The phase space is a finite-dimensional torus. Finite sums of products of one-coordinate functions uniformly approximate its continuous functions, so the product-test formula for the selected law applies.

Split each spatial function into its mean and centered part, then take real and imaginary parts and rescale. It suffices to let each spatial function be either the constant $1$ or a real centered function bounded in absolute value by $1$, and prove that the phase-weighted integral vanishes whenever at least one centered factor occurs.

Finite linear combinations and bounded approximation recover arbitrary bounded tests. Normalize the phase weight to have absolute value at most $1$.

4. Clearing speed denominators leaves only finitely many integer shifts

Source: B p.15, equations (5.3)–(5.4).

Choose a positive integer $h$ such that $e_\alpha=hq_\alpha\in\mathbb N$. In the selected average, write

Fix the initial phases $V$ and $s$. The spatial reading of auxiliary variable $\alpha$ at coordinate $i$ is

The original variable occupies coordinate $b$, with reading

The phase weights do not depend on $n$, and the integer shifts take only finitely many values as $V$ and $s$ vary. This follows because the phases lie in $[0,1)$, we have $s\in[0,h)$, and the finite lists of speeds and indices are fixed.

If factor equivariance holds only modulo null sets at each fixed time, joint measurability and Fubini justify its use in time integrals. Take a common exceptional set over the countably many $n$. Pointwise equivariance simultaneously at all real times is unnecessary.

4.1 Complete blocks and the normalization error

For a selected length $N=c2^k$, set $m=\lfloor N/h\rfloor$. For a test of absolute value at most $1$, replace $[0,N]$ by the first $m$ complete intervals of length $h$ and change the normalization from $1/N$ to $1/(mh)$. The error is at most $2h/N$. For sufficiently large $a$, all selected lengths satisfy $N\ge2^a$, so $m\ge1$ and the error tends uniformly to zero.

The complete part is

Fix $V,s$ below. We need only estimate these $m$-term discrete averages.

5. The original variable and a proper subset of groups

Source: B pp.15–16, the treatment of proper collections of groups in Proposition 5.3.

5.1 A centered auxiliary function is present

Fix a proper subset $J\subsetneq\{1,2,3\}$ of the groups. By (G2), conditional on $V$ these auxiliary variables are mutually independent and independent of $x_0$. The length $N_a$ is a function of $(x_0,u_0)$; given $V$, it depends only on $x_0$.

Suppose at least one auxiliary spatial function is centered. Integrate first over the auxiliary variables and denote the expectation by $R_n(V,s)$. It factors into integrals over the original auxiliary variables $x_\alpha$. Different coordinates on the same $x_\alpha$ have distinct integer speeds $e_\alpha i$.

Here we use the assumed mixing of every finite order: for fixed bounded functions $g_1,\ldots,g_r$, when the integer times separate pairwise,

The time differences within one auxiliary variable diverge as $n$ grows. Since one factor has mean zero, $R_n(V,s)\to0$. There are finitely many shift patterns, so for given $\delta>0$ choose a common $n_0$ such that

holds for every pattern.

Conditional on $x_0,V,s$, the original-variable test has absolute value at most $1$. Even if the length $m=m(x_0,V)$ varies with $x_0$, the absolute value of the test average is at most

After integration, first take $a\to\infty$ and then $\delta\downarrow0$. This gives the required zero limit, controlling both the selected length and its normalization.

5.2 Only the original-variable function is centered

If every auxiliary function equals $1$, only the centered function $f_0$ of the original variable remains. The transformation $T^{hb}$ is mixing and hence ergodic. Birkhoff's theorem gives

almost everywhere, simultaneously for the finitely many shifts $a_0$.

The selected $m(x_0,V)$ tends to infinity at every point, so the same pointwise limit persists. The empty-group case of (G2) says that $x_0$, conditional on $V$, has law $\mu$. Bounded convergence permits integration. This case uses only ordinary mixing and Birkhoff.

This proves every proper-subset group condition involving the original variable in the new system.

6. Length events may alter joint independence of all three groups

Source: B p.16, the start of the all-three-groups argument and equation (5.5).

No original-variable test occurs in (G1). If one group contains no centered factor, only a proper subset of groups is involved, already handled above. We therefore need only consider the case with a centered factor in each of the three groups.

Fix $V,s$. Let $\Omega_j$ be the product probability space of the original auxiliary spatial variables in group $j$. Write the three spatial tests as

These are real functions of absolute value at most $1$. Before restricting to a length event, (G1) gives the product measure on $\Omega_1\times\Omega_2\times\Omega_3$ as their conditional joint law.

6.1 A length event preserves every two-group marginal

For fixed $a$, denote the event $\{N_a=c2^k\}$ by $E_{k,c}^{(a)}$, where $k\ge a$ and $c\in F$. If one length has multiple representations, choose one admissible index so that these events form a partition. Conditional on $V$, write its probability as $p_{k,c}$. These probabilities depend on $a,V$, but for fixed $a$ their sum is $1$.

Conditional on $V$, the event is determined by $x_0$. By (G2), $x_0$ is conditionally independent of any two groups. Thus for any two-group function $K$,

The one-group marginals have the same property, but the full three-group law need not remain a product.

6.2 A subprobability density describes selection

Set $p=p_{k,c}$. Push the joint law of the event and the three groups to the original three-group product space. This gives a subprobability measure of mass $p$, denoted $\nu_E$. For every measurable set $A$,

The right side is the original three-group product measure. Thus $\nu_E$ has a density $r_E$ satisfying

The last inequality follows from $r_E^2\le r_E$. When $p>0$, this is the paper's $r_E=p\rho$. When $p=0$, the subprobability measure vanishes and no conditional law on a zero-probability event is needed.

7. Near orthogonality of the two function sequences

Source: B p.16, the exceptional-graph argument.

Fix an accuracy $\eta>0$. Factor $\langle x_n,x_{n'}\rangle$ over the original auxiliary variables of group 1. For one original variable $x_\alpha$, the times are drawn from

At least one function in this group is centered. If all times that occur are sufficiently far apart, (HM) makes the corresponding integral approach the product of the means and hence zero. Integrals for the other variables have absolute value at most $1$.

Within a single index $n$, the coordinate speeds are distinct, so the time gaps grow with $n$. After discarding finitely many initial indices, only near collisions across the two indices can remain:

Here $i,i',a$ takes finitely many values, and $R$ can be fixed for the given $\eta$ and test functions.

For each fixed $n$, positivity $e_\alpha i'>0$ means that (Near) restricts $n'$ to a real interval of fixed length, containing boundedly many integer choices $n'$. Combine the finitely many obstructions and their reversed pairs into an undirected exceptional graph of finite maximum degree $L$. Outside the graph, $|\langle x_n,x_{n'}\rangle|\le\eta$.

Apply the same argument to $y_n$. The two graphs may differ; take the larger degree bound. Finitely many shift patterns permit common choices of the initial cutoff $n_0$, separation threshold, and degree $L$, independent of $V,s$. They may depend on the fixed tests, speed list, and $\eta$. No mixing rate uniform over all auxiliary systems is assumed.

8. Decomposition along the first $d_m$ covariance directions

Source: B p.17, equations (5.6)–(5.7). Fix one length event, write $m=\lfloor c2^k/h\rfloor$, and suppress the event indices.

Set

In the finite-dimensional span of these $m$ vectors, consider the positive operator

Here $z_n\otimes z_n$ denotes the rank-one operator $u\mapsto\langle u,z_n\rangle z_n$. Let $P_m$ project onto orthogonal eigenvectors of $K_m$ corresponding to its largest $d_m$ eigenvalues. If fewer than $d_m$ eigenvalues are positive, project onto the full range.

8.1 The leading $d_m$ directions: a contribution proportional to $p$

First fix $0<\varepsilon\le1$, then choose $0<\eta\le\varepsilon^2/10^4$. The preceding section supplies near orthogonality. The averaged tensor estimate from the preceding part of Unit 4 yields

where one may take

Fix the tests and $\varepsilon,\eta,L,n_0$ first. After deleting the first $n_0$ terms, retain the rank bound $d_m$ and apply the counting theorem to the remaining $m-n_0$ vectors. The exponential term bounds the number of projection norms exceeding $\varepsilon$, while the remaining terms contribute at most $\varepsilon$ to the average. Deleted terms cost at most $n_0/m$. Since $\sqrt{d_m}\log(2+d_m)=o(\log m)$, we have $\Delta_\varepsilon(m)\to0$. All these norms use the original two-group product measure.

For the event subprobability measure $\nu_E$, the first two groups have marginal $p$ times their original product measure, and the third group has marginal $p$ times its original measure. Cauchy–Schwarz gives

Joint independence of all three groups after restriction to the event is unnecessary. The two-group and one-group marginals suffice. This part of the average contributes at most $p$ times the right side of (Lead).

8.2 The trace bounds the largest remaining eigenvalue

Write $z_n^\perp=(1-P_m)z_n$. Since $\|z_n\|\le1$, the covariance operator satisfies $\operatorname{tr}K_m\le m$. If any directions remain, its largest remaining eigenvalue is at most

Set $A=(\langle z_n^\perp,z_{n'}^\perp\rangle)_{n,n'}$, a positive semidefinite Gram matrix. The two product operators associated with the map $(a_n)\mapsto\sum_na_nz_n^\perp$ are this Gram matrix and the residual covariance operator. They have the same nonzero eigenvalues. Therefore

Also set $B=(\langle w_n,w_{n'}\rangle)_{n,n'}$, which is positive semidefinite and satisfies $\operatorname{tr}B\le m$. Under the original three-group product measure,

The Hilbert spaces are real, so both Gram matrices are symmetric and their trace product equals the displayed double sum. Positive semidefiniteness gives the trace inequality; commutation is not required.

8.3 The square-root probability factor is used only for the remainder

Equations (Density) and (Residual) give

The probability factors differ: $p$ for the leading projection and $\sqrt p$ for the remainder. Bounding both by $\sqrt p$ could lose control when summing over infinitely many length events.

9. Summing over every possible length

Source: B p.17, equation (5.8) and the conclusion of the proof.

For fixed $h$ and finite $F\subseteq[1,2]$, when $k$ is sufficiently large,

uniformly over $c\in F$. Thus

because this is the tail of a convergent series.

For the remainders, Cauchy–Schwarz yields

The probabilities $p_{k,c}$ may depend on $a,V$, but always sum to $1$, so the estimate remains valid.

Sum the leading projection terms with weights $p_{k,c}$. The finitely many shift patterns admit a common maximal cutoff $n_0$ and maximal degree $L$. Define the tail supremum

Since $\Delta_\varepsilon(m)\to0$, we have $\overline\Delta_\varepsilon(M)\to0$, without requiring monotonicity of $\Delta_\varepsilon$. All lengths satisfy $m\ge\lfloor2^a/h\rfloor$, so the sum of the leading contributions is at most

Together with the summable remainder estimate, this also proves absolute summability of the contributions over all length events after decomposition. The number of events need not be finite.

Fix $\varepsilon$ and the resulting $\eta,L,n_0$. First let $a\to\infty$, then $\varepsilon\downarrow0$. The centered test integral involving all three groups tends to zero.

The choice $d_m$ meets two requirements: $\sqrt{d_m}\log(2+d_m)=o(\log m)$ gives a projection count of $o(m)$, while $\sum_kd_{m(k,c)}^{-1}<\infty$ makes the remainder summable along the geometric length classes. The exponent $3/2$ meets both requirements.

10. Measurability, conditioning, and finite iteration

10.1 No parametrized spectral theorem is needed

For each $m$ and integer shift pattern, the covariance operator is finite-dimensional. Make one fixed orthogonal choice in each repeated-eigenvalue space. There are finitely many patterns and countably many $m$. The projections and their function representatives can therefore be chosen in advance and then used according to the measurable pattern and length.

Write the three groups of original variables as $U=(U_1,U_2,U_3)$, with product measure $\kappa=\kappa_1\otimes\kappa_2\otimes\kappa_3$. By (G1), the law of $(V,U)$ is $\nu_V\otimes\kappa$. For every length event, choose a Borel conditional-expectation version

Then, for almost every $v$, the event subprobability measure is $r_{a,k,c}(v,\cdot)\kappa$, with

The common full-measure set must also include the two-group marginal identities. Choose a countable Borel generating algebra for each of the three two-group spaces and first apply (G2) on those sets. Intersect the full-measure sets over the countably many $a,k,c$ and generating-algebra sets. There, each two-group marginal equals $p_{a,k,c}(v)$ times the corresponding product measure. Uniqueness of measures extends the identity to every Borel set. When $p=0$, the subprobability measure is zero and the identities still hold.

The density norm, event probabilities, and all required marginal identities can thus be used simultaneously. The current system and tests are fixed; no union over all auxiliary systems or uncountably many Borel sets is needed. Joint measurability and Fubini allow integration over the phases and time remainder $s$, and restore the bounded phase weight.

The omitted endpoint error tends uniformly to zero and does not change the selected-law limit. Both new laws (G1) and (G2) follow, proving Proposition 5.3 under the stated assumptions.

10.2 From the new maximal distal factor to the old parameter array

Let $P=(\omega_Y(y_i))_{i\in I}$ be the old parameter array. Lemma 3.3, discussed in Part II, places it inside the maximal distal factor of the new flow $M_b(Y)$.

Proposition 5.3 preserves the three-group structure. Lemma 5.2 then gives a product law for all new auxiliary variables conditional on the new maximal distal factor. Condition again on the smaller old parameter $P$. The constant product measure is unchanged, so the variables retain their product law given $P$. This is B Corollary 5.4.

Repeat this argument for finitely many iterations. At each step, fix the system and tests before taking the selected-law limit. No rate of mixing uniform over all auxiliary systems or iteration depths is required.

Guide to the original text and dependencies

• B p.14, Definition 5.1, equations (5.1)–(5.2): the exact conditional laws for the three groups.
• B pp.14–15, Lemma 5.2: extending the conditioning from phases to the maximal distal factor.
• B p.15, Proposition 5.3, equations (5.3)–(5.4): speed denominators, finite shifts, and reduction of the tests.
• B pp.15–16: direct estimates for proper subsets of groups.
• B p.16, equation (5.5): the subprobability density of a length event.
• B p.17, equations (5.6)–(5.8): projection decomposition, Gram matrices, and summation over length classes.
• B pp.17–18, Corollary 5.4: the old parameter array and finite iteration.

The preservation of independence proved here assumes higher-order mixing (HM), the stated relative structure results, the selected-law construction, and the tensor estimate. The proof of (HM) in R is developed separately in the companion reading. This article retains those inputs and does not by itself establish every dependency of the fourfold main theorem.