diff b1f9bc4..56bc728 pointwise_multiple_ergodic_mixing_v2.txt
diff --git a/pointwise_multiple_ergodic_mixing_v2.txt b/pointwise_multiple_ergodic_mixing_v2.txt
index df94c95..05a9fba 100644
--- a/pointwise_multiple_ergodic_mixing_v2.txt
+++ b/pointwise_multiple_ergodic_mixing_v2.txt
@@ -47,16 +47,16 @@
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
+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
+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.
@@ -66,31 +66,31 @@ space.
## 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
-$A,Binmathcal F$calF$
+(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)qquad(|r|to∞).
+ 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
+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 $k,2k,...,nk$.
+x,Tx,... at the n distinct times k,2k,...,nk.
+
-smallskip
noindentMain theorem (Theorem~thm:main below).
-If $T$ is mixing, then for $mu$-almost every $x$
+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 $(X,mathcal F,calF,mu)$
+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
+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.
@@ -98,18 +98,16 @@ functions.
The reason the theorem is delicate is the gulf between two very different
notions of convergence.
-
-
-item Norm convergence asks $A_Ntoint f_1...int f_n$ in $L^2(mu)$.
+ 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.
-item 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
+ 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] 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
@@ -118,7 +116,7 @@ item Pointwise convergence asks that the same limit holds for
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
+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
@@ -137,73 +135,67 @@ verbatim in Ssec:inputs. Everything else is proved here.
The proof is by contradiction, and it has five parts. Read them once before
diving into details.
-
-
-item[(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.
-
-item[(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
+[(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 $Y$.
-
-item[(3)] Channels: a book-keeping device for three groups of outputs.
+ 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
+ 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.
+ 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.
-
-item[(4)] Minimum-support correlation and saturation.
+[(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
+ 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.
-
-item[(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
+ 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 $L^2$
+ 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 $r$.
+ 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
+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: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: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.
-smallskip
+
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.
@@ -216,32 +208,32 @@ check every quantitative estimate.
# Dynamical inputs and standard models
-Fix $nge 2$ and put $I={1,...,n}$. We first record three facts
+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.
+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 $sigma$.
+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
+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.
+functions with respect to sub-sigma-algebras admit bounded Borel versions.
## Input (A): mixing of every finite order
@@ -250,18 +242,18 @@ 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 $t_1,...,t_s$,
+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 $tinZ$.
+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
+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.
@@ -273,36 +265,36 @@ 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:
+varphi on R whose mass is one and whose moments of order 1,2,3 vanish:
[
- int_Rvarphi(s)dd s=1,qquad
+ 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 $varphi$,
+It is standard if, for some constant Gge 1 depending on varphi,
[
- norm{varphi^{(r)}}_∞le (G(r+2))^{G(r+2)}qquad(rge0).
+ 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 $varphi$,
+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 $kto∞$.
+converges almost everywhere as kto∞.
-Moreover, for each $cin[1,2]$ and $ε>0$ there exists a standard
-averaging profile $varphi$ with
+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 $L^1$
+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.
@@ -314,63 +306,63 @@ 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 $a_1,...,a_d$,
+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 $N$.
+converge almost everywhere along all positive integer lengths N.
Proof.
-Add constant-one functions and repeat slopes if $d<3$; both additions
+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
-$gcd(b_1,b_2,b_3)=1$.
+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
+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 $cin[1,2]$.
+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
+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
+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
+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
+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
-$[1,2]$.
+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 $Mle N$,
+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 $etato0$
+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.
@@ -378,22 +370,22 @@ 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
+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 $E[fmid B]=0$,
+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
+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
@@ -403,27 +395,26 @@ invariant probability on the product with the prescribed marginals.
The following hold for standard probability-preserving systems, without
an ergodicity hypothesis.
-
-item 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 $W_Y$.
-item Factors and joinings of distal systems are distal. A distal system is
- relatively distal over each factor. Distality is equivalent for $R$ and any
+ 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.
-item Relative weak mixing passes to nonzero integer powers. The
+ 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.
-item A relatively weakly mixing extension and a relatively distal extension
+ 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.
-item For a distal transformation $R$, every $dge 1$ and every tuple of
- bounded $g_1,...,g_d$, the averages
+ 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
+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.
@@ -432,20 +423,20 @@ 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 $W_Y$.
+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
+The following is the standardised version of L^2 convergence we use
throughout.
-[Relative $L^2$ convergence]
+[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 $G_1,...,G_d$,
+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.
]
@@ -453,10 +444,10 @@ distinct nonzero integers $a_1,...,a_d$ and bounded real $G_1,...,G_d$,
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 $B$.
+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 $hge1$,
+Induction on d. For fixed hge1,
[
F_{i,h}=G_ibar,(G_icirc R^{a_ih}),
]
@@ -464,28 +455,28 @@ 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
+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.
+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
+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.
+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
+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.
@@ -496,47 +487,47 @@ 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 $Nto∞$.
+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
+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.
+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
+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 $n$.
+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
+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,qquad norm{f_i}_∞le 1,
- qquad
+ 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,qquad
+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_*.
@@ -544,24 +535,24 @@ measurable set $E_*⊆ X$ of positive measure with
Set
- C=Xtimes[0,1),qquad m_C=mu ⊗ dd u,
- qquad
+ 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.
+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
+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},qquad
+ 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
+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)}
@@ -572,42 +563,42 @@ there is a continuous nonnegative function $Psi$ on $(R/Z)^I$ with
Proof.
Length-change.
-For a bounded sequence $|H_l|le1$ and positive integers $Lle M$,
+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 $delta$.
+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 $E_*$.
+{(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 $(0,1/(4n))$
+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),
- qquad v=(v_i)_{iin I}in(R/Z)^I.
+ 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 $i$.
+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 $Lge2$,
+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)
@@ -619,138 +610,138 @@ condition, we obtain, uniformly in $z$ and $Lge2$,
The error covers the incomplete last unit interval, the change of index
-summation from $0,...,lfloor Lrfloor-1$ to $1,...,lfloor Lrfloor$,
+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
+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 $L^2(mu)$
+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.
+tends to zero. The uniform error is O(2^{-a}). Taking liminf over
+a completes the proof.
-## Selected laws on every flow over $C$
+## Selected laws on every flow over C
-Fix the selectors and a free ultrafilter $mathcal U$calU$ on $N$.
+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
+[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
+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 $G_i:YtoR$,
+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 $(rho^I)_*lambda_{Y'}=lambda_Y$.
+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.
+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
+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 $barbar G$G$ 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
-$int_Y(1/N_a)int_0^{N_a}G(S_Y^{it}y)dd tdd m_Y(y)toint_YGdd m_Y$.
+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 $lambda_Y$.
+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 $ε$.
+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.
+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.
+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 $pi_{Y'}=pi_Yrho$
+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 $Y'$.
+is identical for Y and Y'.
Inverse limits.
-For $Y_0← Y_1←...$, let $Y_∞$ be the full
+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 $m_{Y_∞}$
+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.
+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
+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),qquad P=(P_i)_{iin I}.
+ 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
-$(Y')^I$.
+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 $W_Y$,
+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 $|J|le 3$,
+For every nonempty J⊆ I with |J|le 3,
law_{lambda_Y}((y_i)_{iin J}mid P)
=⊗_{iin J}m_{Y,P_i}.
@@ -759,48 +750,48 @@ For every nonempty $J⊆ I$ with $|J|le 3$,
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
+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,
+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.
+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
+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
+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.
+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 $s$
+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
+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
+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
+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.
@@ -811,23 +802,23 @@ one chosen slot as the original system.
[Normalized joining flow]
-For $bin I$, define
+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.
+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
+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.
@@ -839,22 +830,22 @@ 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 $C^I$.
+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
+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 $W_C$.
+(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
+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
@@ -866,113 +857,112 @@ cite[Section 5]{FourfoldCompanion}; we spell out every step.
[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
+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
+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,qquad x_gamma=xicirc D_gamma,qquad
+ 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 $m_Y$,
+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,
- qquad H⊊{1,2,3}.
+ 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
+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,
+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 $W_Y$.
+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
+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 $x_0$).
+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 $h$.
+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
+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).
+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
-$Y'$.
+supersystem, gives the stronger conditioning on W_{Y'} for the extension
+Y'.
-## The multiplicative construction $mathsf M_b$sfM_b$
+## 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 $y_bto Y$.
+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:
-
-item the original factor is $x_0(y_b)=xicircpi_Y(y_b)$;
-item the outputs are all $D_gamma(y_i)$, $iin I$, each in the old group
- of $gamma$;
-item the speed of $D_gamma(y_i)$ is $iq_gamma/b$.
+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 $C$.
+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.
+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 $P=(omega_Y(y_i))_{iin I}$.
+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
+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.
@@ -983,9 +973,9 @@ 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
+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.
@@ -994,180 +984,180 @@ 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
+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
+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 $x_0(y_b)$.
+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
+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,qquad e_gamma=hq_gammainN.
+ 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
+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.
+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
+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)
+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
+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.
+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
+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
+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
+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),qquad v_lin L^2(Ω_2),qquad w_lin L^2(Ω_3),
+ 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.
+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
+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
+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.
+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
+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 $eta$;
+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
+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
+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
+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
+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.
+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
+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,qquad z_l=u_l ⊗ v_lquad(0le l<m).
+ 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 $sum_{l<m}z_lz_l^*$,
+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
+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
+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
+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 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 $p$,
+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 $p(ε+o(1))$.
+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
+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.
+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 $sum_{k,c}p_{k,c}(V)=1$,
+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
+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
+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.
+patterns. Letting ato∞ in (eq:selected-law) and then
+εto0 proves the required vanishing.
Both laws in (eq:channel-laws) follow.