Unit 1: From oscillation estimates to pointwise convergence through all lengths

This unit studies §3 of C. Proposition 3.5 is taken as the specified analytic input, and the subsequent convergence argument is developed in detail. The source passages are Definition 3.1 (p.5), Proposition 3.5 (pp.8–9), Lemmas 3.6–3.9 (pp.9–12), and the proof of the main theorem at the end of §3 (pp.12–13).

1. The conclusion, reductions, and specified input

1.1 The conclusion to be deduced

Source: C Theorem 1.1, p.1.

Let $(X,\mathcal F,\mu)$ be a probability space, and let $T$ be invertible, bimeasurable, probability-preserving, and mixing. Fix three pairwise distinct nonzero integers $\lambda_1,\lambda_2,\lambda_3$ and functions $f_1,f_2,f_3\in L^\infty(\mu)$. The paper claims that

almost everywhere, as $N$ runs through all positive integers.

This unit proves that the conclusion follows from the oscillation estimate below. In the paper, that estimate is deduced from C Theorem 2.2 and Lemmas 3.2–3.4. Those results, together with the relevant proofs in H and D, are left for later discussion.

1.2 Normalizing the slopes and functions

Source: the first paragraph of C §2.1, p.3, and the last paragraph of §3, p.13.

Set

Then $b_0,b_1,b_2,b_3$ are pairwise distinct and $\gcd(|b_1|,|b_2|,|b_3|)=1$. Moreover, $T^{\lambda_jn}=S^{b_jn}$, so the averages are unchanged. The transformation $S$ remains invertible and probability-preserving. If $T$ is mixing, then $S$ is mixing, since $|gn|\to\infty$.

If some $f_j$ has $L^\infty$ norm zero, choose the zero representative; the conclusion is immediate. Otherwise, divide each function by its norm. We first use measurable representatives satisfying $|f_j|\le1$ everywhere and restore the constant factors at the end.

The choice of representatives does not change an almost-everywhere conclusion. If $f_j$ and $\widetilde f_j$ differ only on a null set $Z_j$, all discrepancies at integer times lie in

This is a measurable, $T$-invariant null set. We may therefore work with everywhere bounded representatives without assuming completeness of the probability space.

Write

Then $|F_n(x)|\le1$. The analytic input below requires only that $S$ be invertible and measure-preserving. Mixing enters when the limit is identified.

1.3 The precise class of smooth weights

Source: C Definition 3.1, p.5, equations (3.1)–(3.2).

A real-valued function $\varphi$ is called an averaging weight here if $\varphi\in C_c^\infty(\mathbb R)$,

and there is a constant $G\ge1$ such that, for every integer $r\ge0$,

The constant $G$ may depend on $\varphi$. Different approximating weights below may require different values of $G$. No uniform control of their supports or derivatives is asserted.

Such a weight $\varphi$ must change sign. If $\varphi\ge0$, the vanishing second moment would imply $t^2\varphi(t)=0$ almost everywhere. Thus $\varphi$ would be supported almost everywhere on the single point $0$, contradicting its integral being $1$. It cannot be interpreted as a nonnegative probability density.

Define

If $\operatorname{supp}\varphi\subseteq[-B,B]$, with $B\ge1$, each sum is finite and

This is the uniform bound in C (3.10). It applies to one fixed weight; no common bound for all weights is required.

1.4 Proposition 3.5: The assumed oscillation estimate

Source: the statement at the end of C p.8 and its proof on p.9; the conclusion is equation (3.11).

Fix the slopes above and one averaging weight $\varphi$. The paper claims that there is a constant $C_{\varphi,b}\lt\infty$ such that, for every integer $w\ge1$ and every deterministic sequence of integer endpoints

we have

Here $S$ need only be invertible, bimeasurable, and probability-preserving; neither ergodicity nor mixing is required. The functions are normalized by $|f_j|\le1$. The constant may depend on the fixed slopes and weight, but not on the system, functions, endpoints, or number of blocks $w$.

“Deterministic endpoints” means that the same $k_i$ are used for every $x$. The maximizing position may depend on $x$, but each maximum is taken over finitely many scales, so its integrand is measurable. These two facts are compatible.

Estimate (O) is not treated here as an established classical theorem. It is the crucial input claimed by the manuscript and still to be verified. Its stated derivation runs through the local depth-varying estimate C Theorem 2.2 (pp.3–4), the cancellation kernel in C Lemma 3.2 (p.6), the real-line estimate C Lemma 3.3 (p.7), the integer-sequence estimate C Lemma 3.4 (pp.7–8), and finally finite-orbit transference.

2. Why oscillation rules out nonconvergence on a set of positive measure

Source: C Lemma 3.6, pp.9–10. We first formulate the assertion for an abstract sequence of functions.

2.1 The full statement

Let $(Z_k)_{k\ge0}$ be a uniformly bounded sequence of measurable complex-valued functions on a probability space. Suppose there is a constant $C$ such that, for every $w\ge1$ and every deterministic sequence of endpoints $0\le k_1\lt\cdots\lt k_{w+1}$,

Then $Z_k(x)$ converges on a measurable set of full measure. The conclusion specifies neither the limit nor a rate of convergence.

2.2 Tail diameters and a fixed amplitude

Source: p.9, the first paragraph in the proof of Lemma 3.6 and the tail-diameter formula.

Set

Each supremum involves only countably many measurable functions, so $\omega_K$ is measurable. It decreases with $K$ and is finite by uniform boundedness; hence its limit exists and is measurable. The condition $\omega(x)=0$ is equivalent to $(Z_k(x))$ being Cauchy. By completeness of the complex numbers, this is also equivalent to convergence.

Suppose, to the contrary, that $\mu\{\omega>0\}>0$. Since

some $m$ yields a set of positive measure on the right. Write $\delta=1/m$ and

From now on, $E,d,\delta$ are fixed. Individual points may originally have different tail amplitudes; this choice restricts them to a common positive amplitude.

2.3 A deterministic finite block from an infinite tail

Source: p.10, the opening of the proof of Lemma 3.6, immediately after the definition of $E$.

Fix any integer $K\ge0$. For $p,q\ge K$, the triangle inequality implies

Taking suprema, we obtain

Set

As $L\to\infty$, the quantities $M_{K,L}$ increase to the tail supremum above. The monotone convergence theorem implies

There is therefore a deterministic integer $L>K$ such that

The integer $L$ is chosen using the integral; it is not a function of $x$. Estimate (B) does not require every point of $E$ to make an excursion within this block, or require the excursions to occur at the same scale. It only requires the integral of the block maximum to reach a fixed positive value.

2.4 Linear lower growth versus a square-root upper bound

Source: p.10, the last three sentences of the proof of Lemma 3.6.

Start with $k_1=0$. At each step, use the previous endpoint as the new $K$ and choose the next endpoint by (B). This produces deterministic, strictly increasing endpoints, with the same lower bound $\delta d/2$ on every block. Consequently, for every $w$,

Thus $\sqrt w\le2C/(\delta d)$. The right side is fixed, whereas $w$ can be arbitrarily large, a contradiction. Hence $\omega=0$ almost everywhere, which proves the assertion.

Apply this to $Z_k=V_k^\varphi$, using (O) and the uniform bound from §1.3. The smooth averages for each fixed weight converge almost everywhere. Mixing has not been used.

2.5 Why sublinear growth matters

A bound of only $Cw$ would not suffice. For example, $Z_k(x)=(-1)^k$ is independent of $x$. Its maximal oscillation on each integer block with distinct endpoints is $2$, so the total is $2w$, but the sequence does not converge. What matters is sublinear growth uniformly over all choices of endpoints.

Even if each point continues to make excursions, no single excursion time need work for every point. Taking a finite maximum before integrating avoids this error in the order of quantifiers.

3. Multiple $L^2$ convergence from ordinary mixing

Source: C Lemmas 3.7–3.8, pp.10–11. We state the norm-convergence theorem for any finite number of factors. In the present triple-average application, the transformation is $S$.

Let $(X,\mathcal F,\mu)$ be a probability space, and let $T$ be invertible, bimeasurable, probability-preserving, and mixing. Fix an integer $d\ge1$, pairwise distinct nonzero integers $a_1,\ldots,a_d$, and functions $f_1,\ldots,f_d\in L^\infty(\mu)$. Then

This is a norm-convergence statement. It does not directly yield almost-everywhere convergence through all lengths, and it is distinct from higher-order mixing without time averaging.

3.1 From mixing of sets to correlations of functions

Source: C p.10, the beginning of §3.3 and equation (3.13).

Mixing of sets asserts that, for fixed $E,F\in\mathcal F$, we have $\mu(E\cap T^{-m}F)\to\mu(E)\mu(F)$ as $|m|\to\infty$. We need to deduce that, for fixed bounded measurable functions $g,h$,

If $g,h$ are simple functions, express them as finite linear combinations of indicator functions. The integral on the left expands into finitely many set correlations, and the conclusion follows term by term from the definition.

In general, for each $\varepsilon>0$, choose simple functions $g_\varepsilon,h_\varepsilon$ with $\|g-g_\varepsilon\|_\infty\le\varepsilon$ and $\|h-h_\varepsilon\|_\infty\le\varepsilon$. Such uniform approximations follow by dividing the bounded complex ranges into finitely many small cells. For every $m$,

The difference between the corresponding products of integrals has the same upper bound. Fix $\varepsilon$, apply the simple-function conclusion as $|m|\to\infty$, and then let $\varepsilon\downarrow0$. This proves the required function formulation. Complex coefficients cause no change in the argument.

3.2 Lemma 3.7: A finite Hilbert-space inequality

Source: C p.10, the proof from the zero-extension identity through the estimate for finite $H$.

Let $v_n$ belong to a Hilbert space, with $\|v_n\|\le M$. Suppose that, for every fixed positive integer $h$, the limit

exists. The inner product is taken to be linear in the first variable. For each fixed integer $H\ge1$, we have

To prove this, fix $H$ and take $N\ge H$. For this finite estimate only, extend the sequence outside $1\le n\le N$ by setting $v_n$ to zero. Then

By the Cauchy–Schwarz inequality,

Expand the square on the right. The diagonal terms sum to at most $HNM^2$, and the terms with displacement $h$ occur $H-h$ times. Thus

With $H$ fixed, let $N\to\infty$ to obtain (H). If $\gamma_h\to0$, then

Letting $H\to\infty$ now shows that the average vectors converge to zero in norm.

3.3 Lemma 3.8: Induction on the number of factors

Source: C p.11, the proof of (3.14), ending with the expansion after subtracting means.

First consider the case in which at least one function has mean zero. Set

For fixed $h$, invariance of the measure implies

If $d=1$, the right side is independent of $n$, so $\gamma_h=\int_Xq_{1,h}\,\mathrm{d}\mu$.

Suppose the theorem holds for $d-1$ functions. The slopes $a_j-a_1$, for $2\le j\le d$, are pairwise distinct and nonzero. The induction hypothesis therefore implies

Denote the left side by $G_{N,h}$ and its constant limit by $c_h$. Since $q_{1,h}\in L^2(\mu)$, the Cauchy–Schwarz inequality implies

Insert this limit into (C). For fixed $h$, deleting the final $h$ terms introduces an error of $O(h/N)$. We obtain

Since $a_j\ne0$, mixing implies

At least one $f_j$ has mean zero, so $\gamma_h\to0$. By (H), we have $N^{-1}\sum_{n=1}^Nv_n\to0$ in $L^2$.

Finally, write each function as

Expand the product. The term consisting entirely of constants is $\prod_jm_j$. Every other term contains at least one mean-zero function, and there are only finitely many such terms. Their averages therefore tend to zero in $L^2$. This completes the induction.

3.4 Assumptions and order of limits

• Distinct slopes ensure that the difference slopes in the induction are nonzero.
• Each slope is nonzero, so $|a_jh|\to\infty$ and mixing applies.
• Invertibility permits negative times and the measure-preserving change of variables.
• Boundedness places every product in $L^2$ and controls the finitely many terms deleted for fixed $h$.
• First fix $h$, then let $N\to\infty$. In (H), first fix $H$ and only at the end let $H\to\infty$. No rate of mixing is needed.

The slope conditions cannot be dropped. For instance, with two equal slopes and $f_2=\overline{f_1}$, the average is an ordinary ergodic average of $|f_1|^2$, whose limit is generally not $|\int f_1|^2$. If a zero slope is allowed and the other function is identically one, the zero-slope observable retains its dependence on the starting point.

4. Smooth weighted $L^2$ convergence and identification of the pointwise limit

Source: C Lemma 3.8, p.11, from the positive- and negative-time partial sums through (3.15), followed by the Fatou argument.

4.1 Positive and negative times must be treated separately

Return to the notation $F_n=\prod_{j=1}^{3}f_j\circ S^{b_jn}$ and $p=\prod_j\int f_j\,\mathrm{d}\mu$ from §1. Apply the norm-convergence theorem separately to the slopes $b_j$ and $-b_j$. This yields

Both sets of slopes are pairwise distinct and nonzero, so both applications are valid. Negative times cannot be omitted here, since the averaging weight may have support on the negative half-line.

More generally, fix any $\chi\in C_c^1(\mathbb R)$ with $\int\chi=1$. We will prove that

This conclusion requires neither the first three moments to vanish nor the higher-derivative bounds of §1.3. Those two conditions belong to the smaller class of weights used for the oscillation estimate.

4.2 The finite summation-by-parts identity

Choose $B\ge1$ such that $\operatorname{supp}\chi\subseteq[-B,B]$. Set

Since $M_a/a>B$, we have $w_{M_a}^\pm=0$, and all subsequent weights also vanish. With the convention $R_0^\pm=0$,

Expanding the terms and shifting indices yields the exact identity

The terminal term $w_{M_a}^\pm R_{M_a}^\pm$ vanishes, and the initial term contains $R_0^\pm=0$, so there are no remaining boundary terms.

By the mean value theorem,

Thus the positive- and negative-time parts each satisfy

4.3 Why the bound is $o(1)$

Given $\varepsilon>0$, first choose a fixed $N_0$ such that, for both signs and every $n\ge N_0$, we have $\|R_n^\pm\|_2\le\varepsilon n$. This is the only use of the $o(N)$ estimate.

Split the right side of (A) into $n<N_0$ and $n\ge N_0$. The first part is a fixed finite sum multiplied by $a^{-2}$, so it tends to zero as $a\to\infty$. The second part is at most

Since $M_a/a\to B$, its limit superior is at most $\varepsilon B^2\|\chi'\|_\infty/2$. Finally let $\varepsilon\downarrow0$. Both weighted errors tend to zero. No rate of decay of $\|R_n^\pm\|_2/n$ has been assumed.

The centered term at $n=0$ is $a^{-1}\chi(0)(F_0-p)$, whose $L^2$ norm is $O(a^{-1})$. The constant part satisfies

because $\chi$ is continuous and compactly supported, and its uniform-grid Riemann sums converge. Combining the three parts proves (W).

4.4 Identifying the existing pointwise limit

Take $\chi=\varphi$ and $a=2^k$. Then $V_k^\varphi\to p$ in $L^2$. Section 2 already shows that $V_k^\varphi$ converges on a measurable full-measure set $E_\varphi$; denote its limit by $L_\varphi$. Fatou's lemma implies

Hence $L_\varphi=p$ almost everywhere. If necessary, remove one more measurable null set from $E_\varphi$ so that $V_k^\varphi(x)\to p$ for every $x\in E_\varphi$.

Dominated convergence is not needed to obtain the integral conclusion; Fatou's lemma suffices. Mixing establishes the norm limit in this section, while the oscillation estimate established the existence of the pointwise limit through all scales.

5. Smooth weights: derivative bounds and approximation with vanishing moments

Source: the smooth-function construction following C Definition 3.1, p.5, and Lemma 3.9, p.12. These passages address the derivative condition and the moment correction, respectively.

5.1 A nonnegative smooth function with the required derivative growth

Set

This function is smooth and positive on $(-1,1)$. We must check that its zero extension is smooth at the endpoints and that all derivatives obey the growth condition of §1.3.

Fix $|t|<1$ and write $d=1-t^2$. In the complex plane, consider $|z-t|\le d/16$. Then

Consequently, $\operatorname{Re}(1-z^2)\ge3d/4$ and $|1-z^2|\le5d/4$. Using $\operatorname{Re}(1/q)=\operatorname{Re}(q)/|q|^2$, we obtain

The denominator does not vanish in the disk, so $\exp(-1/(1-z^2))$ is analytic there, with absolute value at most $\exp(-12/(25d))$. The Cauchy derivative estimate therefore yields

For fixed $r$, the right side tends to zero as $d\downarrow0$, since exponential decay dominates every fixed power. Each interior derivative extends continuously to zero at both endpoints. Checking the derivatives successively shows that the zero extension belongs to $C_c^\infty(\mathbb R)$.

For a bound in terms of $r$, set $u=1/d\ge1$. The maximum of $u^r\mathrm e^{-cu}$ on the positive half-line is $(r/(c\mathrm e))^r$; for $r=0$, use a constant bound separately. By (G), there is an absolute constant $C$ such that

Since $r!\le(r+1)^r$, choosing a sufficiently large fixed $G$ bounds the right side by $[G(r+2)]^{G(r+2)}$. Normalize by setting

Then $\rho$ is nonnegative, has mass $1$, is supported in $[-1,1]$, and still satisfies a derivative bound of the same form.

The function $\rho$ is only an ingredient in constructing an averaging weight; it does not yet have the required vanishing moments. Its nonnegativity does not conflict with the fact that an averaging weight must change sign.

5.2 Stability of the derivative class under fixed operations

Source: C p.5, the paragraph on closure properties following Definition 3.1.

A fixed translation leaves the derivative supremum norms unchanged. For a fixed scale $a>0$ and a fixed center $z$,

A fixed dilation therefore introduces a factor growing at most exponentially in $r$, which can be absorbed by increasing $G$. Differentiation of a fixed order replaces $r$ by $r+m$. A fixed finite linear combination adds only finitely many derivative bounds, again absorbed by increasing $G$. The word “fixed” is important: no single $G$ has been obtained uniformly over all translation parameters, scales, or linear combinations.

5.3 The full statement of Lemma 3.9

For every $c\in[1,2]$ and $\varepsilon>0$, there is an averaging weight $\varphi$ in the sense of §1.3 such that

The weight $\varphi$ must simultaneously have mass $1$, vanishing first, second, and third moments, compact support, and the stated derivative growth. Density of $C_c^\infty$ in $L^1$ alone does not prove this assertion.

5.4 Smoothing the interval weight and fixing the initial moments

Source: C p.12, the first paragraph of the proof of Lemma 3.9.

Set $\rho_\delta(t)=\delta^{-1}\rho(t/\delta)$ and

The function $v$ is real-valued, nonnegative, smooth, and compactly supported, with $\int v=1$ and support contained in $[-\delta,c+\delta]$. For any real $s$, the symmetric difference of two length-$c$ intervals whose positions differ by that amount has length $2\min(|s|,c)$. Hence

By the integral triangle inequality and $\rho_\delta\ge0$,

Choose $\delta$ small enough that $\|v-h_c\|_1<\varepsilon/2$. Since $\|h_c\|_1=1$, we also have

The preceding subsection shows that $v$ belongs to the required derivative class. Now fix $v$ and write

In general, $m_j\ne0$, so these moments still need correction. Subsequent constants may depend on the function $v$, which is now fixed.

5.5 Invertibility of the moment-correction matrix

Source: C p.12, the paragraph in the proof of Lemma 3.9 defining $P_j$, $\rho_{i,A}$, and the matrix $M$.

Fix four pairwise distinct real numbers $z_0,z_1,z_2,z_3$, for example $0,1,2,3$. Set

Thus $P_j$ is a monic polynomial of degree $j$. Index the rows of the matrix by $j$ and the columns by $i$, and set

If

then $M=LV$. The matrix $L$ is lower triangular, with every diagonal entry equal to $\mu_0=1$. Thus

This identity makes explicit the triangular change from a Vandermonde matrix mentioned in the paper. The matrix $M$ is now a fixed real invertible matrix.

5.6 Moving a small amount of mass to a large scale

For a scale $A\ge1$, define

This is nonnegative and has mass $1$. The substitution $t=A(z_i+u)$ yields

Solve the linear system

Since $A\ge1$, the vector $\ell^1$ norm satisfies

Define

Row zero of the system states that $\sum_i\beta_i=0$, so $\int\varphi=1$. For $j=1,2,3$, multiply row $j$ by $A^j$ to obtain

Since each $\rho_{i,A}$ is nonnegative and has mass $1$,

Choose $A$ large enough that the right side of (M) is less than $\varepsilon/2$. Then

All coefficients are real, and $A$ is now fixed. The weight $\varphi$ is a finite sum of compactly supported smooth functions; §5.2 ensures the derivative growth condition. All requirements in the definition of an averaging weight have therefore been checked.

5.7 Why the expanding support matters

The support of $\rho_{i,A}$ lies in $A[z_i-1,z_i+1]$. Higher accuracy may require larger $A$, and hence larger supports for the correction terms. Their $L^1$ mass is $O(A^{-1})$, whereas their $j$-th moments may be amplified by $A^j$. A small $L^1$ change can therefore correct the low-order moments exactly.

If instead every approximant were required to be supported in the same interval $[-R,R]$, then

For $R\ge c$, this would force the second moments to converge as the $L^1$ error tends to zero. But $\int t^2\varphi=0$, whereas $\int t^2h_c=c^2/3>0$, a contradiction. A common fixed support cannot be added to the approximation statement. The paper does not impose one.

6. From smooth dyadic scales to all integer lengths

Source: C pp.12–13, the proof of the main theorem following Lemma 3.9, especially (3.16)–(3.17) and the finite rational grid on p.13.

6.1 One countable intersection of full-measure sets

For each rational $c\in[1,2]$ and each integer $h\ge1$, choose in advance an averaging weight $\varphi_{c,h}$ such that

By §§2 and 4, for each fixed pair $(c,h)$ there is a measurable full-measure set $E_{c,h}$ on which

The index set $([1,2]\cap\mathbb Q)\times\mathbb N$ is countable, so

is still measurable and has full measure. From now on, fix $x\in E$. All rational scale ratios and all approximation accuracies are handled on the same set. No uncountable intersection over functions or ratios is taken.

6.2 The exact endpoint identity for interval weights

Set $N_k=\lfloor c2^k\rfloor$. Since $h_c=c^{-1}\mathbf1_{(0,c]}$, the sum includes exactly the integers $1\le n\le N_k$. Thus

Since $|A_{N_k}|\le1$ and $0\le c2^k-N_k<1$, the left side differs from $A_{N_k}$ by at most $(c2^k)^{-1}$ in absolute value. For any fixed averaging weight $\varphi$, the triangle inequality now yields the paper's equation (3.16):

The function $|h_c-\varphi|$ is bounded and compactly supported, and its only possible discontinuities are at $0,c$. It is therefore Riemann integrable, and the uniform-grid sum converges to $\|h_c-\varphi\|_1$. Take $\varphi=\varphi_{c,h}$, fix $h$, and let $k\to\infty$. We obtain

Now let $h\to\infty$. Then

The weight is fixed before the scale tends to infinity. No uniform estimate is asserted for derivative constants, support radii, or oscillation constants that vary with the approximation accuracy.

6.3 Comparing two averaging lengths

Let $(a_j)$ be a complex sequence with $|a_j|\le1$. For $1\le n\le N$,

Estimating the two terms separately yields

This inequality holds pointwise; it uses neither measure preservation nor mixing. It controls averages at the previously untreated lengths between neighboring grid points.

6.4 Fix the grid, take the scale limit, then refine the grid

Fix an integer $s\ge1$ and take the finite rational grid

For any $N\ge1$, choose the unique integer $k\ge0$ such that $2^k\le N<2^{k+1}$. Let $i$ be the largest index satisfying

The index $i=0$ always qualifies. The index $i=s$ corresponds to $2^{k+1}>N$, so $i<s$. By maximality and the distance between adjacent grid points,

By (L),

By (Q), the averages at each fixed grid point converge to $p$. Since the grid is finite, the maximum of their errors also tends to zero:

Combining these estimates, we obtain