% ============================================================================
% Pointwise Multiple Ergodic Averages for Mixing Transformations
% A self-contained, complete proof, written to be read from front to back.
%
% The file is plain ASCII holding LaTeX source, same convention as the
% original. Every theorem, proposition, and lemma is proved here. Three
% companion results (listed as INPUT A, B, C below) are taken from other
% preprints in this series; they are stated precisely and used as black
% boxes, exactly as in the original.
%
% Reading keys, so nothing is hidden:
%
%   (X,F,mu,T)      T is invertible, bimeasurable and mu-preserving.
%   mixing          mu(A cap T^{-r}B) -> mu(A)mu(B) as |r| -> infty.
%   L^infty         bounded real measurable functions (complex by linearity).
%   L^2, ip, norm   the usual Hilbert space of mu-square-integrable functions.
%   A_N             A_N(f_1,...,f_n)(x) = (1/N) sum_{k=1}^N prod_j f_j(T^{jk}x).
%
% The single result proved below is Theorem 0.1: on every mixing system,
% A_N converges almost everywhere to the product of the integrals.
% ============================================================================























# Pointwise Multiple Ergodic Averages for Mixing Transformations

maketitle

Abstract: 
Let T be an invertible mixing probability-preserving transformation
of a probability space (X,mathcal F,calF,mu).  For every integer nge 2
and every tuple f_1,...,f_nin L^∞(mu), the consecutive
multiple ergodic averages
[
 A_N(f_1,...,f_n)(x) = frac1Nsum_{k=1}^Nprod_{j=1}^n f_j(T^{jk}x)
]
converge mu-almost everywhere to prod_{j=1}^nint_X f_jddmu as
Nto∞ through all positive integers.  No rate of mixing is
assumed, and (X,mathcal F,calF,mu) need not be a standard probability
space.


# Introduction


## The problem and the answer

noindentSetting.
(X,mathcal F,calF,mu) is a probability space, T:Xto X is invertible,
bimeasurable and measure preserving.  T is mixing if for every
A,Binmathcal FcalF

 mu(Acap T^{-r}B)→mu(A)mu(B)(|r|to∞).

Fix nge 2 and f_1,...,f_nin L^∞(mu).  The object of study
is the sequence of consecutive multiple ergodic averages
[
 A_N(f_1,...,f_n)(x)=frac1Nsum_{k=1}^Nprod_{j=1}^n f_j(T^{jk}x).
]
These average the integrand along the finite arithmetic progression
x,Tx,... at the n distinct times k,2k,...,nk.


noindentMain theorem (Theorem~thm:main below).
If T is mixing, then for mu-almost every x
[
 A_N(f_1,...,f_n)(x)→prod_{j=1}^nint_X f_jddmu
]
as Nto∞ through all positive integers.  The space (X,mathcal F,calF,mu)
need not be standard and no rate of mixing is assumed.

The exceptional null set depends on the tuple (f_1,...,f_n) and on
the system, but not on N.  The theorem says nothing about averages
taken over a different set of slopes, and nothing about uniformity over
functions.

## Why this is nontrivial

The reason the theorem is delicate is the gulf between two very different
notions of convergence.
 Norm convergence asks A_Ntoint f_1...int f_n in L^2(mu).
  Host--Kra [cite] (Theorem 1.1) and Ziegler [cite]
  (Corollary 1.8) proved this for every finite length on every probability-preserving
  system.
 Pointwise convergence asks that the same limit holds for
  mu-almost every x.  It is not a corollary of norm convergence:
  bounded sequences can converge in L^2 and still oscillate on a set of
  positive measure.  Bourgain's double recurrence theorem
  [cite] proved n=2; higher lengths have only been settled
  under structural hypotheses: K-systems
  [cite], weakly mixing systems whose Pinsker
  component has singular spectrum [cite], and weakly mixing
  systems whose pairwise independent self-joinings are independent
  [cite].  Huang--Shao--Ye
  [cite] proved the pointwise statement for every finite
  length on ergodic measure-distal systems.

Theorem~thm:main extends all of these: it covers every mixing
system, every length nge 2, and does not need ergodicity or standardness
beyond the reductions we make at the start.

The paper continues the fourfold mixing theorem of
[cite].  Three companion results are used.  (A) Ordinary
mixing implies mixing of every finite order, the assertion of Rokhlin's
multiple-mixing problem [cite].  (B) An oscillation
estimate for triple averages at distinct integer slopes on any invertible
probability-preserving system [cite], which yields
pointwise convergence of three-term averages at distinct positive integer
slopes on any system.  (C) The Furstenberg--Zimmer structure facts and
relative disjointness between relatively weakly mixing and relatively
distal extensions [cite].  All three are quoted
verbatim in Ssec:inputs.  Everything else is proved here.

## Proof strategy: five steps

The proof is by contradiction, and it has five parts.  Read them once before
diving into details.
[(1)] Assume failure: a positive witness at variable lengths.
  If Theorem~thm:main fails for the tuple (f_1,...,f_n), there is a
  smaller index set J_0⊆ {1,...,n}, centered real functions f_i
  bounded by 1, a set E_* of positive measure and delta>0, such that
  B_L(x)=(1/L)sum_{l=1}^Lprod_{iin J_0}f_i(T^{il}x) exceeds 2delta for
  arbitrarily large integers L, at each xin E_*.  This is Ssec:selected.
[(2)] Turn the failure into a probability law.
  Move to the roof-one suspension C=Xtimes[0,1).  Choose a measurable
  selector N_a at each starting point, say N_a takes values in a finite
  dyadic grid, and choose it so the trajectory average of the witness exceeds
  delta on E_* for arbitrarily large a.  Along any free ultrafilter
  on N, this defines a probability law lambda_Y on Y^I for every flow
  Yto C.  lambda_Y has fixed marginals, is flow invariant, comes with
  small-marginal laws for at most three slots, and is functorial in Y.
[(3)] Channels: a book-keeping device for three groups of outputs.
  A quantitative dependence on the original point is captured by three
  groups of outputs, i.e.~factor maps D_gamma:Yto C.  The
  two conditions are: the outputs have product spatial law given their phases,
  and the original spatial coordinate has product law jointly with any proper
  collection of groups.  Two facts are needed.  Conditioning can be strengthened
  to conditioning on W_Y, the maximal measure-distal factor.  And the
  construction Yrightsquigarrowmathsf M_b(Y)sfM_b(Y)
  (each slot gets its own copy of Y, scaled by 1/b) is a channel.
  The second fact is where tensor packing enters; it is the technical heart.
[(4)] Minimum-support correlation and saturation.
  Among all nonzero correlations of the form
  E_{lambda_Y}[Phi(P)prod_{iin J}h_i(L_i(y_i))], over all channels Y,
  choose one with the minimum number k of slots; among those, maximize the
  number r of slots whose role is output-only.  Then upgrade to a
  saturated channel: no further extension of the channel can improve the
  conditional expectation of any test given the parameters sigma(P), or the
  parameters together with one slot sigma(P,y_l).  A short repackaging turns
  the support-k correlation on the saturated channel into a correlation
  supported on k slots of the regrouped channel
  Z=mathsf M_b(Y),sfM_b(Y), where one slot flips from input to output.
[(5)] Square argument and contradiction.
  On the selected law of Z we have an Itimes I array.  Row b and column
  b both have the old law lambda_Y; they meet at a common corner.  Saturation
  gives a row--column identity for a bilinear pairing of two conditional
  expectations.  A Hilbert-space lemma on independent random vectors in L^2
  of a conditional fibre forces the pairing not to vanish.  Packaging it back
  as a correlation on Z produces a correlation of support k with r+1
  output-only slots, contradicting the maximality of r.


Only input (B) is used at length 3 on an arbitrary system.  Everything for
length nge 4 is either norm-convergence (L^2) or pointwise convergence on
distal systems.

## Map of the paper

{ll}
Ssec:inputs     & standard models, and inputs (A)--(C) stated precisely.
Ssec:selected   & the failure witness, the suspension, the selectors, the laws lambda_Y.
Ssec:channels   & channels, channel conditioning, channel closure by tensor packing.
Ssec:correlations& minimum support, saturation, the regrouped channel Z=mathsf M_b(Y).sfM_b(Y).
Ssec:square     & the square argument and the contradiction.
Ssec:packing    & appendix: the tensor packing theorem.



A remark on what is and is not elementary.  The proof uses genuine results
of ergodic theory (Furstenberg--Zimmer structure, Khintchine-style matrix
estimates, sated-extension constructions).  We do not pretend otherwise.
What we have tried to do is to lay out each step so that, on a first reading,
the reader can see why it is being done; and, on a second reading, can
check every quantitative estimate.

% The remaining sections follow.

# Dynamical inputs and standard models


Fix nge 2 and put I={1,...,n}.  We first record three facts
from other papers, and set up the standard-model reduction.  The pointwise
input on an arbitrary system is available only for three slots;
for any finite number of slots we have only L^2 convergence.

## Reduction to a standard probability space

Let (X,mathcal F,calF,mu,T) be the fixed system, and f_1,...,f_n bounded
real functions.  Record all integer translates of the f_j as coordinates
in a single map Phi:Xto K^{Ztimes{1,...,n}}, where K=[-M,M] is
a compact real interval containing the ranges of all f_j.  The pushforward
nu=Phi_*mu is a Borel probability on a compact metrizable space, and T
pushes forward to the invertible shift sigma on K^{Ztimes I}, preserving
nu.  Since T is mixing, so is sigma.

The original functions become coordinate functions on this factor.  A
countable union of translated null sets runs through the change of
representatives, so mu-almost-everywhere convergence of the pushed-forward
averages pulls back to the same statement on X.  From now on the
system is standard: (X,mathcal F,calF,mu,T) is a standard Borel probability
space with an invertible measure-preserving transformation.

Throughout the proof, factors and conditional laws are understood modulo null
sets; every function is taken to have a Borel representative.  For a pair of
standard probability spaces, conditional expectations of bounded Borel
functions with respect to sub-sigma-algebras admit bounded Borel versions.

## Input (A): mixing of every finite order

This is the content of Rokhlin's multiple-mixing problem for one
transformation.

[INPUT A: higher-order mixing; cite[Theorem 1.1]{MixingCompanion}]

Let T be invertible and mixing on a probability space.  For every integer
sge 2, every g_1,...,g_sin L^∞(mu) and every s-tuple of
integers t_1,...,t_s,
[
 int_Xprod_{j=1}^s g_j(T^{t_j}x)ddmu(x)→prod_{j=1}^sint_X g_jddmu
]
as min_{ine j}|t_i-t_j|to∞.  For s=1 the identity holds at
every single time tinZ.


The cited theorem is stated for sets and for ordered times.  Simple-function
approximation, invariance of mu under T and the finitely many orderings
of the times give the stated formulation.  No rate of mixing is asserted or
used.

## Input (B): smooth triple averages with no mixing

The second input treats averages of at most three functions, but at distinct
integer slopes, and on any invertible probability-preserving
system, not just a mixing one.

[Averaging profile]
An averaging profile is a real compactly supported smooth function
varphi on R whose mass is one and whose moments of order 1,2,3 vanish:
[
 int_Rvarphi(s)dd s=1,
 int_R s^rvarphi(s)dd s=0quad(r=1,2,3).
]
It is standard if, for some constant Gge 1 depending on varphi,
[
 norm{varphi^{(r)}}_∞le (G(r+2))^{G(r+2)}(rge0).
]


[INPUT B: smooth triple convergence; cite[Prop 3.5, Lem 3.6, Lem 3.9]{SlopesCompanion}]

Let R be any invertible probability-preserving transformation of a
standard probability space.  Let b_1,b_2,b_3 be distinct nonzero integers
with greatest common divisor one, and let g_1,g_2,g_3 be bounded real
functions.  Then for every standard averaging profile varphi,
[
 V_k^varphi(y)=sum_{linZ}2^{-k}varphi(l/2^k)prod_{j=1}^3 g_j(R^{b_jl}y)
]
converges almost everywhere as kto∞.

Moreover, for each cin[1,2] and ε>0 there exists a standard
averaging profile varphi with
[
 norm{varphi-c^{-1}mathbf 1_bf1_{(0,c]}}_{L^1(R)}<ε.
]


The second assertion approximates the uniform density on (0,c] in L^1
by a smooth profile with the same mass and moments.  Neither part uses
mixing or ergodicity.

## From smooth to ordinary length

Combining INPUT B with INPUT A gives pointwise convergence for three-slot
averages at distinct positive integer slopes, along all lengths.  This is
the only point where any pointwise input for arbitrary systems is used.

[Triple pointwise convergence on every system]

Let R be any invertible probability-preserving transformation of a
standard probability space.  For bounded real functions g_1,...,g_d
with dle 3, and distinct positive integer slopes a_1,...,a_d,
the averages
[
 frac1Nsum_{l=1}^Nprod_{j=1}^dg_j(R^{a_jl}y)
]
converge almost everywhere along all positive integer lengths N.


Proof. 
Add constant-one functions and repeat slopes if d<3; both additions
multiply the average by a constant and change nothing about its
convergence.  Write the resulting slopes as g,b_1,g,b_2,g,b_3 where
gge 1 and b_1,b_2,b_3 are distinct positive integers with
gcd(b_1,b_2,b_3)=1.

By scaling we may assume |g_j|le1.  Put S=R^g and
[
 F_l=prod_{j=1}^3 g_j(S^{b_j l}y).
]
Then |F_l|le 1.  Let N_k=lfloor c2^krfloor for cin[1,2].

The smooth-to-ordinary step.
For varphi a profile, we have
[
 V_k^varphi(y)=sum_{linZ}2^{-k}varphi(l/2^k)F_l.
]
For h_c=c^{-1}mathbf 1_bf1_{(0,c]} set
[
 W_k(y)=sum_{linZ}2^{-k}h_c(l/2^k)F_l.
]
Then W_k=(1/N_k)sum_{l=1}^{N_k}F_l+O(1/(c2^k)) because h_c is the
normalized indicator of (0,c], using lfloor c2^krfloor=N_k and the
error on the last c2^k-N_k terms being at most 1/(c2^k).  Also
[
 |V_k^varphi(y)-W_k(y)|le 2^{-k}sum_{linZ}|h_c(l/2^k)-varphi(l/2^k)|.
]
By Riemann sums the right side tends to norm{h_c-varphi}_1 uniformly in
y.  Now pick a sequence of profiles varphi^{(m)} with
norm{h_c-varphi^{(m)}}_1to0.  On a common conull set (intersection
of the conull sets supplied by Proposition~prop:input-smooth at each
m) all V_k^{varphi^{(m)}}(y) converge.  For mto∞ the above
display and the triangle inequality show that W_k(y) is Cauchy.  Since
W_k and the ordinary average differ by O(2^{-k}), the ordinary averages
along N_k=lfloor c2^krfloor form a Cauchy sequence for mu-almost
every y.  This holds for every c in a countable dense subset of
[1,2].

All lengths, not just dyadic.
For any bounded sequence |H_l|le1 and positive integers Mle N,
[
 |frac1Msum_{l=1}^M H_l-frac1Nsum_{l=1}^N H_l|le(2|N-M|)/(max(N,M)).
]
For 2^kle N<2^{k+1} pick a finite rational grid cin[1,2] with mesh
less than some eta, and compare N to the sequence values at
lfloor c2^krfloor.  The error is at most O(eta).  Letting etato0
on a countable union of scales preserves almost everywhere convergence.
Therefore the average along every complex of lengths converges almost
everywhere.

Identifying the limit.
Lemma~lem:relative-norm applied over the trivial factor shows that
the ordinary three-slot averages converge to the product of the means in
L^2.  If their almost-everywhere limit exists on a positive-measure set,
it must equal that L^2 limit, so all the a.e. limits we produce at
different c agree.  This identification is not needed below; only
existence of the a.e. limit is.


## Input (C): Furstenberg--Zimmer structure facts

The third input packages the standard relative ergodic theory.  Recall an
extension Yto B of an invertible system R is relatively weakly
mixing if for bounded f,g with E[fmid B]=0,

 frac1Hsum_{h=1}^Hnorm{E[barbar ff,(gcirc R^h)mid B]}_2^2→0.

It is relatively compact if the L^∞(B)-modules spanned by
finite sums sum_j a_jF_j (a_jin L^∞(B), F_jin L^2 fixed) are
dense.  A relatively distal extension is a tower of relatively
compact extensions and inverse limits.  A system is distal if it is
distal over the trivial factor.  A joining of two systems is an
invariant probability on the product with the prescribed marginals.

[INPUT C: relative structure; cite[Prop 2.2]{FourfoldCompanion}]

The following hold for standard probability-preserving systems, without
an ergodicity hypothesis.
 Every system Y has a maximal measure-distal factor W_Y, and the
  extension Yto W_Y is relatively weakly mixing.  Distal factors of any
  extension pull into W_Y.  Automorphisms commuting with R preserve W_Y.
 Factors and joinings of distal systems are distal.  A distal system is
  relatively distal over each factor.  Distality is equivalent for R and any
  nonzero integer power.
 Relative weak mixing passes to nonzero integer powers.  The
  conditional product of relatively weakly mixing fibre laws over an invariant
  joining of the base is relatively weakly mixing over that joining.
 A relatively weakly mixing extension and a relatively distal extension
  of the same base are relatively disjoint: every invariant joining identifying
  their bases equals the conditional product over that base.
 For a distal transformation R, every dge 1 and every tuple of
  bounded g_1,...,g_d, the averages
  [
   frac1Nsum_{l=1}^Nprod_{j=1}^dg_j(R^{jl}y)
  ]
  converge almost everywhere.

For a flow Y (a jointly Borel action of R), the maximal distal factor of
time one is preserved by the flow, and its action at every nonzero rational
time is distal.


The cited results include the nonergodic form of the Furstenberg--Zimmer
structure theorem [cite] and the pointwise theorem
for distal systems cite[Theorem C]{HuangShaoYe2019}; see
cite[Prop 2.2]{FourfoldCompanion} for the proof in the nonergodic setup.  In
the flow assertion, invariance of W_Y under every flow time is immediate
because each time commutes with time one.  If r=p/qne 0 is rational, the
q-th power of S_Y^r equals S_Y^p, which is distal; hence S_Y^r is
distal on W_Y.

## Norm convergence on relatively weakly mixing extensions

The following is the standardised version of L^2 convergence we use
throughout.

[Relative L^2 convergence]

Suppose Yto B is a relatively weakly mixing extension of R.  For
distinct nonzero integers a_1,...,a_d and bounded real G_1,...,G_d,
[
 left|frac1Nsum_{l=1}^N(prod_{i=1}^dG_icirc R^{a_il}-prod_{i=1}^dE[G_imid B]circ R^{a_il})right|_2→0.
]


Proof. 
By telescoping and scaling, it suffices to show the average of
v_l=prod_i G_icirc R^{a_il} tends to zero in L^2 when |G_i|le1 for
all i and one of them, say G_s, is centered over B.

Induction on d.  For fixed hge1,
[
 F_{i,h}=G_ibar,(G_icirc R^{a_ih}),
]
so by invariance
[
 ip{v_l}{v_{l+h}}=int_Y F_{s,h}prod_{ine s}F_{i,h}circ R^{(a_i-a_s)l}dd m_Y.
]
The induction hypothesis replaces the average over l of the product over
ine s by its conditional expectation over B.  The remaining product is
B-measurable and bounded by one; hence
[
 limsup_{Nto∞}|frac1Nsum_{l=1}^Nip{v_l}{v_{l+h}}|lenorm{E[F_{s,h}mid B]}_1.
]
For d=1 this bound holds directly; for d>1 it is the induction output.
The Hilbert-space van der Corput inequality bounds the size of the limit in
L^2 by a constant times
[
 frac1H+frac1Hsum_{h=1}^Hnorm{E[F_{s,h}mid B]}_1.
]
Relative weak mixing of B⊆ Y over R^{a_s} (Assertion~3 of
Proposition~prop:input-structure) makes each norm{E[F_{s,h}mid B]}_1
small in average, and Cauchy--Schwarz (since F_{s,h} is bounded) makes the
whole expression tend to zero as Hto∞.  This completes the induction.


Inserting constant-one functions in omitted positions, and using the
pointwise theorem for distal systems (Assertion~5 of
Proposition~prop:input-structure) to get an L^2 limit of the
distal-factor averages, gives an L^2 limit for arbitrary bounded tuples
at distinct positive integer slopes on any standard system.  On the
original mixing system, the trivial factor already satisfies
(eq:rwm), and the limit is the product of the means.

## A first corollary for the original system

Before we start the proof, we note the easy consequence we will use to
produce the failure witness in Ssec:selected.


Suppose T is mixing.  For any f_1,...,f_nin L^∞(mu) with
int f_iddmu=0 for some i, the averages A_N(f_1,...,f_n) tend to
zero in L^2(mu) as Nto∞.


Proof. 
Apply Lemma~lem:relative-norm with B the trivial factor, the
identity transformation R=T, and a_i=i.  Because T is mixing, the
trivial factor is a relatively weakly mixing extension.  Then the product
prod_iE[f_imid B]=prod_iint f_i=0, and the lemma follows.


# The failure witness and selected laws


Assume, toward a contradiction, that Theorem~thm:main fails for the
fixed slopes I={1,...,n} and some f_1,...,f_nin L^∞(mu).
We extract a positive witness, move to the roof-one suspension C, define
measurable averaging lengths N_a, and along a free ultrafilter obtain a
probability law lambda_Y on Y^I for every flow Y over C.  The
construction follows cite[Sections 3--6]{FourfoldCompanion}, with the
extension and conditioning facts spelled out for every n.

## A positive witness at selected lengths

Expanding each f_j into its mean and centered part, and each centered part
into real and imaginary parts, we find a nonempty J_0⊆ I and real
functions f_i, iin J_0, with
[
 int_Xf_i,ddmu=0, norm{f_i}_∞le 1,
 
 B_L(x)=frac1Lsum_{l=1}^Lprod_{iin J_0}f_i(T^{il}x)
]
such that B_L does not tend to zero on a set of positive measure.  Here and
below B_L has integer subscript.  Lemma~lem:norm-vanishes gives
B_Lto0 in L^2(mu), so the failure is genuinely pointwise: B_L
oscillates on a set of positive measure, and its limit is 0 in L^2.
After changing the sign of one f_i, there exist delta>0 and a
measurable set E_*⊆ X of positive measure with

 B_L(x)>2deltaquadfor arbitrarily large L,
 xin E_*.


## The suspension and the selectors

Set

 C=Xtimes[0,1), m_C=mu ⊗ dd u,
 
 S_C^t(x,u)=(T^{lfloor u+trfloor}x,{u+t}),

and write xi(x,u)=x, υ(x,u)=u, viewing υ also as a
coordinate on R/Z.  Then (C,m_C,S_C) is a measure-preserving flow.

[Selected positive witness]

There are a finite set F⊆[1,2] with 1in F, and measurable
functions N_a:Cto[2^a,∞), ainN, each independent of the
phase coordinate, with

 N_a(x,u)in{c2^k:cin F, kge a},
 N_a(x,u)=2^a (xnotin E_*),

and such that B_{lfloor N_a(x,u)rfloor}(x)>delta on E_*.  Moreover
there is a continuous nonnegative function Psi on (R/Z)^I with

 liminf_{ato∞}int_Cfrac1{N_a(z)}
 int_0^{N_a(z)}
 Psi((υ(S_C^{it}z))_{iin I})
 prod_{iin J_0}f_i(xi(S_C^{it}z))dd tdd m_C(z)>0.



Proof. 
Length-change.
For a bounded sequence |H_l|le1 and positive integers Lle M,
[
 |frac1Lsum_{l=1}^LH_l-frac1Msum_{l=1}^MH_l|
 le(2|M-L|)/(max(M,L)).
]
Pick a finite grid F⊆[1,2] containing 1, so fine that for every
integer L there exist cin F and k=lfloorlog_2Lrfloor with
|L-c2^k|<eta L; choose eta so small that the length-change bound
stays below delta.  Then for every xin E_* and every L at which
B_L(x)>2delta, the corresponding dyadic average
B_{lfloor c2^krfloor}(x) exceeds delta.

The measurable choice.
Fix an enumeration of the countable set
{(c,k):cin F,,kinN}.  For ainN and xin E_*, let (c,k) be
the first pair in this enumeration with kge a and
B_{lfloor c2^krfloor}(x)>delta.  On E_* we set N_a(x,u)=c2^k;
on X∖ E_* we set N_a=2^a.  As a map Cto[2^a,∞) this is
measurable and independent of u.  By construction
B_{lfloor N_a(x,u)rfloor}(x)>delta on E_*.

The phase witness.
Choose continuous nonnegative circle functions alpha,beta, not
identically zero, supported inside (1/4,1/2) and (0,1/(4n))
respectively.  Put

 Psi(v)=alpha(2v_1-v_2)beta(v_2-v_1),
  v=(v_i)_{iin I}in(R/Z)^I.

For z=(x,u) and t=l+s with linZ, 0le s<1, the phase of slot i
is υ(S_C^{it}z)={u+is}.  The support of Psi then forces
0<u+is<1 for every iin I.  In particular, on the support of Psi,
xi(S_C^{it}z)=T^{il}x for every i.

Summing complete unit intervals of length 1 in t and using the support
condition, we obtain, uniformly in z and Lge2,

 &frac1Lint_0^L
 Psi((υ(S_C^{it}z))_i)
 prod_{iin J_0}f_i(xi(S_C^{it}z))dd t
 notag
 &hspace{18mm}
 =alpha(u)(int_0^1beta(s)dd s)B_{lfloor Lrfloor}(x)
 +O(L^{-1}).
 

The error covers the incomplete last unit interval, the change of index
summation from 0,...,lfloor Lrfloor-1 to 1,...,lfloor Lrfloor,
and the normalisation.

Integrating.
On E_*, with L=N_a(z), the integral of the main term in
(eq:suspension-witness) is at least
[
 deltamu(E_*)(int_0^1alpha(u)dd u)(int_0^1beta(s)dd s)>0,
]
since B_{lfloor N_a(z)rfloor}>delta on E_*.  On X∖ E_*, the
function N_aequiv 2^a is independent of u, and B_{2^a}to0 in L^2(mu)
by Lemma~lem:norm-vanishes; hence the contribution of this region
tends to zero.  The uniform error is O(2^{-a}).  Taking liminf over
a completes the proof.


## Selected laws on every flow over C

Fix the selectors and a free ultrafilter mathcal UcalU on N.

[Flow over C; extensions]
A flow over C is a standard probability space (Y,m_Y) with a
jointly Borel probability-preserving action S_Y^t, tinR, and a
probability-preserving factor map pi_Y:Yto C with pi_YS_Y^t=S_C^tpi_Y
mod m_Y for each fixed t.  An extension rho:Y'to Y of such
flows is a probability-preserving factor map, equivariant modulo null sets,
with pi_{Y'}=pi_Ycircrho modulo null sets.  We do not require a common
conull set on which every real-time factor identity holds; joint
measurability and Fubini justify their uses in time integrals.


[Selected joining laws]

Every flow Y over C has a unique probability lambda_Y on Y^I
satisfying, for all bounded measurable G_i:YtoR,

 int_{Y^I}prod_{iin I}G_i(y_i)ddlambda_Y
 =lim_{atomathcal U}calUint_Yfrac1{N_a(pi_Yy)}
 int_0^{N_a(pi_Yy)}prod_{iin I}G_i(S_Y^{it}y)dd tdd m_Y(y).

Its marginals are m_Y, and lambda_Y is invariant under
(y_i)_{iin I}mapsto(S_Y^{it}y_i)_{iin I} for every tinR.  If
rho:Y'to Y is an extension over C, then (rho^I)_*lambda_{Y'}=lambda_Y.

Countable inverse limits exist in this category.  The selected law at the
limit projects to the selected law at each stage.  Pullbacks of the stage
sigma-algebras have dense union in L^2(lambda_Y) at the limit.


Proof. 
Construction at each a.
For each a, the expression before the ultralimit in (eq:selected-law)
defines a probability measure lambda_{Y,a} on Y^I, since N_acircpi_Y
is measurable, ymapsto S_Y^{it}y is jointly Borel, and the integrand is
bounded.

Marginals.
We show lambda_{Y,a}to m_Y on marginals as ato∞ along every
bounded G:YtoR.  By Birkhoff applied to the auxiliary function
ymapstoint_0^1G(S_Y^{is}y)dd s and to S_Y^i, the continuous-time
averages (1/L)int_0^LG(S_Y^{it}y)dd t converge almost everywhere as
Lto∞, and their a.e. limit barbar GG satisfies
int_Ybarbar GGdd m_Y=int_YGdd m_Y since every deterministic average has
this integral.  As N_age 2^ato∞, bounded convergence gives
int_Y(1/N_a)int_0^{N_a}G(S_Y^{it}y)dd tdd m_Y(y)toint_YGdd m_Y.

Ultralimit.
Embed Y into a compact metrizable K (as a Borel subset with m_Y
extended by zero).  Along the ultrafilter mathcal U,calU,
lambda_{Y,a} against continuous functions on K^I converges to a
positive normalised functional, hence a probability K^I.  Its marginals
are m_Y, so it is concentrated on Y^I.  Denote it lambda_Y.

Formula for all bounded tests.
For bounded G_i:YtoR with |G_i|le1, approximate each by continuous
functions on K with the same bound in L^1(m_Y) to within ε.
The difference of the two product integrals (eq:selected-law) at stage
a is bounded by sum_iε.  Under lambda_Y these errors are
L^1-errors, and under lambda_{Y,a} they converge to those errors by
marginal convergence.  Letting εto0 gives the formula.
Product tests determine a unique probability on Y^I, giving uniqueness.
All identities depend only on G_i up to a.e. equivalence.

Invariance.
Fix t_0inR and shift the time interval in (eq:selected-law) by
t_0.  The selector N_acircpi_Y does not change under
S_Y^{t_0}.  For |G_i|le1 the error is at most 2|t_0|2^{-a},
proving invariance under S_Y^{t_0} in each slot.

Functoriality.
Substitute G_icircrho for G_i in the formula and use pi_{Y'}=pi_Yrho
plus Fubini and equivariance.  The right-hand side of (eq:selected-law)
is identical for Y and Y'.

Inverse limits.
For Y_0← Y_1←..., let Y_∞ be the full
countable product of the underlying standard spaces with the consistent
inverse-limit measure and coordinatewise flow.  Joint measurability is
automatic for coordinatewise actions of R on a countable product.  The
projections are measurable, S_{Y_∞}^t preserves m_{Y_∞}
because every cylinder has the same measure as its preimage.  Stages are
factor maps mod null sets; projection through Y_0 gives the factor
Y_∞to C.  Functoriality gives the claimed projections of stage
selected laws.  The stage sigma-algebras generate the product
sigma-algebra; pullbacks are increasing and generate L^2(lambda_Y),
by the usual L^2 approximation by increasing sigma-algebras.


## Distal parameters and small marginals

For a flow Y over C, let omega_Y:Yto W_Y be its maximal measure-distal
factor for time one, and nu_Y=(omega_Y)_*m_Y.  Write m_{Y,w} for
its conditional measures, so m_Y=int_{W_Y}m_{Y,w}ddnu_Y(w).  Under
lambda_Y put

 P_i=omega_Y(y_i), P=(P_i)_{iin I}.

By Assertion 1 of Proposition~prop:input-structure, W_Y is preserved
by every flow time.  Under an extension rho:Y'to Y, the pullback of W_Y
is a distal factor of Y' for time one, hence is contained in W_{Y'}.
Thus P on Y^I is a measurable function of the corresponding array on
(Y')^I.

[Parameter and small-marginal laws]

For every flow Y over C and every bounded measurable H_i on W_Y,

 E_{lambda_Y}prod_{iin I}H_i(P_i)
 =lim_{Lto∞}int_Yfrac1Lint_0^L
 prod_{iin I}H_i(omega_Y(S_Y^{it}y))dd tdd m_Y(y).

For every nonempty J⊆ I with |J|le 3,

 law_{lambda_Y}((y_i)_{iin J}mid P)
 =⊗_{iin J}m_{Y,P_i}.



Proof. 
From discrete to continuous time.
Write t=l+s with lge0 integer and 0le s<1.  For each fixed s, a
continuous-time product becomes the discrete-time product at slopes i with
slot functions G_icirc S_Y^{is}, evaluated at times lge0.  If the
discrete averages converge almost everywhere, Fubini and bounded convergence
give that the s-average of them does.  Starting the discrete average at
l=0 rather than l=1, and dropping an incomplete final unit interval,
contributes an error that tends to zero.  The same argument applied to real
Lge2 in place of dyadic N shows convergence along all real lengths.

Deterministic parameters.
Apply this reduction to the functions H_icircomega_Y, which are fixed by
all flow times (Assertion 1 of Proposition~prop:input-structure);
pointwise convergence on distal systems follows from Assertion 5.  Since
N_ato∞, bounded convergence with (eq:selected-law) gives
(eq:deterministic-parameters).

Small marginals off the parameter.
The extension Yto W_Y is relatively weakly mixing.  For |J|le3,
Lemma~lem:triple applied to the discrete averages at slopes iin J
shows that the J-marginal attains the integrated deterministic-length law.
By Lemma~lem:relative-norm, the difference of the deterministic
average prod_{iin J}G_i(y_i) and the same product with each G_i
replaced by E[G_imid W_Y] tends to zero in L^2.  Integrating in s
gives

 E_{lambda_Y}prod_{iin J}G_i(y_i)
 =E_{lambda_Y}prod_{iin J}(E_{m_Y}[G_imid W_Y](y_i)).


Small marginals given P.
Multiplying G_i by arbitrary bounded test functions of P_i gives

 law_{lambda_Y}((y_i)_{iin J}mid(P_i)_{iin J})
 =⊗_{iin J}m_{Y,P_i}.

Now upgrade from (P_i)_{iin J} to the full array P.  Under
S_Y^{it} acting on slot i, the J-marginal system of
(eq:small-own-parameters) is relatively weakly mixing over its own
parameter law, by Assertions 2 and 3 of
Proposition~prop:input-structure.  The full P-system is a joining of
the distal systems W_Y at integer slopes i and is thereby distal.  Its
joint law under lambda_Y is an invariant joining identifying that common
subarray; by relative disjointness (Assertion 4) the joining splits, which is
exactly the assertion.  This completes the proof.


The full parameter law is unaffected by the selectors, even though the full
state law need not be.  We turn the state law into a new flow while preserving
one chosen slot as the original system.

[Normalized joining flow]

For bin I, define

 mathsf M_b(Y)=(Y^I,sfM_b(Y)=(Y^I,lambda_Y,
   S_{mathsf M_b(Y)}sfM_b(Y)^t(y_i)_{iin I}=(S_Y^{it/b}y_i)_{iin I}).

This is a standard probability-preserving flow over C via
(y_i)_{iin I}mapstopi_Y(y_b); the projection onto slot b is an
extension onto Y.  The array P belongs to the maximal measure-distal
factor of mathsf M_b(Y)sfM_b(Y) for time one.


Proof. 
Invariance follows from Proposition~prop:selected-law after rescaling
the time parameter by 1/b.  The factor map is pi_Ycirc p_b.  Slot b
has speed 1 and marginal m_Y, giving (y_i)_imapsto y_b as a factor
onto Y.  In slot i, the W_Y-factor is distal for time i/b, by
Assertion 2 (equivalence for rational powers); the law of P is an
invariant joining of these rational-time distal systems, hence distal.
Maximality of the distal factor gives the last assertion.


[The retained positive correlation]

With the functions and phase test of Lemma~lem:selected-witness,

 int_{C^I}Psi((υ(z_i))_i)
 prod_{iin J_0}f_i(xi(z_i))ddlambda_C>0.

The phase array is a function of the distal parameter array of C^I.


Proof. 
Continuous functions on the compact product of circles are uniformly
approximable by finite sums of products of continuous one-coordinate
functions.  Apply this to Psi, use (eq:selected-law) for each
resulting product, then apply the strictly positive lower bound in
(eq:selected-positive-limit).  The phase at time zero on C is
distal, hence lies in W_C.


# Channels: three groups of outputs


The selected length is determined by the original point in C.  It
therefore can change the joint law of other coordinates, even when their
unconditioned law is a product.  To control the dependence we pack the
outputs into three groups.  Independence of the original spatial coordinate
from any two groups will turn out to hold automatically, and this is enough
to cancel the effect of selecting the length.  The construction adapts
cite[Section 5]{FourfoldCompanion}; we spell out every step.

## Channels

[Channel]

A channel is a flow Y over C with factor pi_Y:Yto C, together
with a finite list of probability-preserving maps D_gamma:Yto C called
outputs, and positive rational numbers q_gamma, satisfying

 D_gamma(S_Y^ty)=S_C^{q_gamma t}D_gamma(y)

for each fixed t, mod m_Y.  Partition the output indices into three
groups, numbered 1,2,3; empty groups are allowed.  Set
[
 x_0=xicircpi_Y, x_gamma=xicirc D_gamma,
 d_Y=(x_gamma)_gamma,
]
and for H⊆{1,2,3} let d_{Y,H} be the sublist of d_Y from
groups in H.  Let V be the joint of the phase coordinates of pi_Y
and all D_gamma (i.e. V=(υcircpi_Y,(υcirc D_gamma)_gamma)).
Require, under m_Y,

 
  law(d_Ymid V)&=⊗_gammamu,
  law(x_0,d_{Y,H}mid V)&=
    mu ⊗ ⊗_{gamma in groups H}mu,
     H⊊{1,2,3}.
 

An empty spatial list has the one-point law.


Thus the spatial outputs have a full product law independent of the phases.
The original spatial coordinate x_0 has product law jointly with the
outputs in any proper collection of groups; independence from all
three groups is not required.  The suspension C, with no outputs,
is a channel.

## Conditioning and extensions

The first observation is that the phase variable V in Definition~def:channel
can be replaced by the maximal measure-distal factor W_Y.

[Conditioning and extensions]

For a channel Y, both identities in (eq:channel-laws) hold with
conditioning on W_Y in place of V.  Every flow extension of Y over
C is a channel with the pulled-back output data, and the same
strengthening holds there.


Proof. 
Choose a positive integer h such that hq_gammainZ for every output.
The phase vector V is a factor fixed by time h, and its time-one action
is distal, so V belongs to W_Y.  For any spatial tuple appearing in
(eq:channel-laws), its joint factor with V has product law over
V; the joint factor with W_Y is a joining of this over V.

At time h, the spatial coordinates evolve by a product of positive
integer powers of T (the powers are hq_gammainN and h for x_0).
This product is mixing by INPUT A.  The assertion follows first for
product tests, by mixing of each power, and then for all bounded tests by
approximation.  Consequently the spatial factor is relatively weakly
mixing over V for time h.

On the other hand, W_Y is distal for time h and relatively distal over
V.  Relative disjointness (Assertion 4 of
Proposition~prop:input-structure) makes the spatial factor and
W_Y relatively independent over V, so its conditional spatial law given
W_Y is the product law in (eq:channel-laws).

Every flow extension preserves the joint laws of the original data, so the
channel identities pull back.  The same argument, applied with a common
supersystem, gives the stronger conditioning on W_{Y'} for the extension
Y'.


## The multiplicative construction mathsf M_bsfM_b

Recall from Lemma~lem:normalized-joining that for bin I,
mathsf M_b(Y)sfM_b(Y) has state space Y^I, measure lambda_Y, flow
(y_i)_{iin I}mapsto(S_Y^{it/b}y_i)_{iin I}, and factor y_bto Y.

[Channel closure]

Let Y be a channel and bin I.  Define data on Z=mathsf M_b(Y)sfM_b(Y) by:
 the original factor is x_0(y_b)=xicircpi_Y(y_b);
 the outputs are all D_gamma(y_i), iin I, each in the old group
  of gamma;
 the speed of D_gamma(y_i) is iq_gamma/b.

Then these data make Z a channel over C.


The proof is the technical heart of the paper, and is given in
Ssubsec:closure-proof.  It uses INPUT A (mixing of every order) at
length n and the tensor packing theorem stated in Ssec:packing.

[Copied output law]

For every channel Y, the copied spatial list
(d_Y(y_i))_{iin I} under lambda_Y has its full product law
conditional on P=(omega_Y(y_i))_{iin I}.


Proof. 
Apply Proposition~prop:channel-closure and
Lemma~lem:channel-conditioning to mathsf M_b(Y).sfM_b(Y).  By
Lemma~lem:normalized-joining, the old parameter array P belongs to
the maximal distal factor of mathsf M_b(Y).sfM_b(Y).  Conditioning the
constant product law (eq:channel-laws) down to P proves the
assertion.


## Proof of channel closure


This is Ssubsec:closure-proof, the proof of
Proposition~prop:channel-closure.

Proof. [Proof of Proposition~prop:channel-closure]
Each new output D_gamma(y_i) is probability-preserving and has positive
rational speed iq_gamma/b, so the equivariance (eq:output-equivariance)
holds with q=iq_gamma/b.  It remains to prove the two conditional laws in
(eq:channel-laws).

Reductions.
Test the laws with products of bounded spatial functions and products of
bounded phase-test functions.  Such tests determine the laws, and complexity
is unchanged by approximation.  Expanding each spatial function into its
mean and centered part, and using centered functions of bounded absolute
value, reduces the proof to real spatial factors that are either constants or
centered and bounded by 1.  We may bound every phase test by 1; the
assertion below is linear in the phase test and so is preserved.

A test containing a spatial factor at x_0(y_b) uses outputs from only a
proper subset of the groups; we treat such tests inside the "proper set of
groups" paragraph.  Consequently we may assume that in the "full output
list" case, no spatial test involves x_0(y_b).

Time intervals and offsets.
Use the selected-law formula (eq:selected-law) with base flow being
Y, and condition on the old phase vector V under m_Y.  Fix an integer
h>0 clearing all output speed denominators, and write t=hl+s with
lge0 an integer and 0le s<h.  For fixed V,s, all phase weights in
the test are independent of l.  The spatial function for output gamma
at slot i is evaluated at

 T^{e_gamma i l+a_{gamma,i}}x_gamma, e_gamma=hq_gammainN.

The offsets a_{gamma,i} depend on V,s and range over a fixed finite
set, by (eq:suspension).  If the original-factor test is present, its
argument is T^{hbl+a_0}x_0, with the same finite-offset property.
Equivariance and Fubini justify these formulas inside the time integrals.

On an event where the selected length is N=c2^k (cin F, kge a), put
m=lfloor c2^k/hrfloor.  Replacing the normalized time integral by the
average over m complete intervals, followed by integration against
ds/h, has error at most 2h2^{-a}.  For all sufficiently large a,
mge1 on each admissible event.

Case 1: a proper set of groups (or all outputs 1).
Suppose some proper subset H⊊{1,2,3} is used together with
x_0.  Given V, the relevant old output coordinates have product law
and are independent of x_0, by (eq:channel-laws).  The selector
depends on V,x_0 only.  Suppose at least one output occurrence is centered.
For fixed V,s,x_0, integrate the spatial output product at index l.
The expectation factors over the old output indices gamma.  For any
index with a centered occurrence, the times (eq:channel-times) in
distinct slots separate as lto∞.  Mixing of all orders (INPUT A)
makes the coordinate integral tend to zero (each factor bounded, one
centered so the product law mean is 0).  Convergence is uniform over the
finitely many offset patterns and finitely many V-sections.  A Ces`aro
limit therefore gives zero for the full average.  All other factors have
absolute value le1, so bounded convergence completes this case.

If, on the other hand, every output test is 1, the only centered factor
is x_0 (bounded by 1 and int x_0ddmu=0).  Birkhoff's theorem for
T^{hb} applied to x_0 gives
[
 frac1msum_{l<m}x_0(T^{hbl+a_0}x_0)→0
]
for mu-almost every x_0, simultaneously for the finitely many offsets
a_0.  Since N_ato∞ pointwise, the selected lengths share this
limit.  The conditional law of x_0 given V is mu, so bounded
convergence completes this case.

Case 2: full output list with a centered occurrence per group.
It remains to treat the full output list, with no original-factor test.  If
one group has no centered occurrence, every spatial test in that group is
1, and the preceding argument applies.  Thus suppose each group contains
a centered occurrence.  For fixed V,s, write the spatial product at index
l as
[
 u_lin L^2(Ω_1), v_lin L^2(Ω_2), w_lin L^2(Ω_3),
]
where Ω_j carries the product measure with one copy of mu for each
old output in group j.  These are real unit-vectors (functions bounded by
1).  Given V, the old spatial output law is the product of these three
spaces.  The vectors depend on V,s only through the finite offset patterns.

Almost orthogonality with a bounded-degree graph.
For every eta>0, after omitting finitely many initial indices, the
families (u_l) and (v_l) each have absolute pair inner products at most
eta outside a graph of bounded maximum degree L_0.  All bounds are
uniform in the offset patterns.  To see this, in each of the two groups
choose a centered occurrence at some output index gamma.  Its coordinate
integral in a pair inner product is small by INPUT A whenever the times in
the two copies of (eq:channel-times) are sufficiently separated.  The
times within each copy separate once l,l' exceed a fixed threshold.
Failure of separation between copies requires one of finitely many
inequalities
[
 |e_gamma i l-e_gamma i'l'+a'|le R.
]
For fixed l, each inequality permits only a bounded number of integers
l', because e_gamma,i'>0; and symmetrically.  Their union gives a
bounded-degree graph.  Coordinate integrals for other output indices are
bounded by 1.  The separation threshold and L_0 may depend on eta;
no rate of mixing is needed.

Conditioning on a selected length.
For fixed selector index a, partition into the events N_a=c2^k,
cin F, kge a, choosing one representation when a length has more
than one.  Given V, write their probabilities as p_{k,c}(V).  On one
such event of probability p>0, the conditional spatial law of any two
groups, and of each single group, is unchanged.  Indeed the event is
determined by V,x_0, which is independent of any proper set of groups by
(eq:channel-laws).  The full three-group law may change.  Its
density rho relative to the product law satisfies

 0le prhole1,qquadint prho=p,qquadnorm{prho}_2lesqrt p.

The bound 0le prhole1 is the absolute-continuity statement; the second
identity is the normalisation; the third is Cauchy--Schwarz.  These hold
for almost every V, and the length events are countable.

The rank bound.
The rank parameter must meet two requirements: the packing bound must be
o(m), and the reciprocal bounds 1/d_m must be summable over dyadic
length scales.  For m=lfloor c2^k/hrfloor set

 d_m=lceil(1+log m)^{3/2}rceil, z_l=u_l ⊗  v_lquad(0le l<m).

Let Pi_m be the orthogonal projection onto up to d_m leading
eigenvectors of the positive covariance operator sum_{l<m}z_lz_l^*,
taking its full range if its rank is smaller.

Fix 0<εle1 and then fix 0<etaleε^2/10^4 in the
preceding graph bound.  Theorem~thm:packing of the Appendix gives

 frac1msum_{l<m}norm{Pi_mz_l}leε+o(1).

Here o(1) tends to 0 uniformly over the offset patterns.  Indeed, after
the fixed initial indices are removed, the number of projection norms at
least ε is at most
[
 C(1+L_0)ε^{-2}
 exp!(Cε^{-2}√(d_m)log(2+d_m))=o(m).
]
The removed indices contribute o(1).  Crucially, ε,eta and
L_0 are fixed before mto∞.

The projected terms.
Cauchy--Schwarz under the conditional event law uses its unchanged
two-group and one-group marginals.  Then, term by term, including the event
probability p,
[
 |pint(Pi_mz_l)(omega_1,omega_2),w_l(omega_3),rho(omega)ddomega|
 le pnorm{Pi_mz_l}_2norm{w_l}_2
 le pnorm{Pi_mz_l}_2.
]
Thus the projected contributions total at most p(ε+o(1)).

The residual vectors.
For the residual vectors z_l'=(1-Pi_m)z_l, set
A=(ip{z_l'}{z_{l'}'})_{l,l'<m} and B=(ip{w_l}{w_{l'}})_{l,l'<m}.
The covariance trace is at most m, so the remaining eigenvalues of the
covariance operator, and hence the nonzero eigenvalues of A, are at most
m/d_m.  If the rank is at most d_m, the residual is zero.  Also, B is
positive semidefinite with trace at most m.  Therefore
[
 left|frac1msum_{l<m}z_l' ⊗  w_lright|^2
 =(tr(AB))/(m^2)lefrac1{d_m}.
]
By (eq:channel-density), the residual contribution, including the
event probability, is at most √(p/d_m).  Multiplication by the fixed
phase test of absolute value at most 1 does not increase either bound.

Summing over lengths.
Write m(k,c)=lfloor c2^k/hrfloor.  For fixed h this is comparable to
2^k uniformly in cin F.  Since sum_{k,c}p_{k,c}(V)=1,
Cauchy--Schwarz gives

 sum_{kge a, cin F}sqrt{frac{p_{k,c}(V)}{d_{m(k,c)}}}
 le
 (sum_{kge a, cin F}frac1{d_{m(k,c)}})^{1/2}→0.

The series converges because its summands are O(k^{-3/2}) thanks to the
rank bound (eq:channel-dimension).  The projected contributions have
total limsupleε, by (eq:channel-projected-average) and
the same probability identity.  These estimates are uniform in V,s and may
be integrated.  There is no measurable-choice issue for the projections:
choose one for each of the countably many sizes and finitely many offset
patterns.  Letting ato∞ in (eq:selected-law) and then
εto0 proves the required vanishing.

Both laws in (eq:channel-laws) follow.