Unit 2: From local estimates to oscillation on finite orbits

Unit 1 took the oscillation bound in C Proposition 3.5 as an input. This unit traces it back to C Theorem 2.2, examining the proofs of Lemmas 3.2–3.4 and Proposition 3.5. The source passages are C pp.3–4 and 6–9.

Theorem 2.2 remains a specified hypothesis here. Its proof occupies §§4–6 of the paper and invokes central analytic results from H and D. This unit checks the deduction of the oscillation bound from that local estimate.

1. What the local estimate permits

1.1 Slopes, intervals, and depth

Fix pairwise distinct integers $b_0=0,b_1,b_2,b_3$ satisfying

Take a translated dyadic grid of half-open intervals. A finite tree rooted at $J$ contains every ancestor between each selected interval and the root. For an interval $I$ in the tree, its depth is

Greater depth means a smaller interval. Write $d$ for the maximal depth of the tree.

1.2 The complete admissibility conditions for kernels

Source: C Definition 2.1, p.3, equations (2.1)–(2.3).

Fix $\delta_0>0$ and $C_0\ge1$. On a length-$r$ interval $I$, the kernel $w_I\in C_c^\infty(\mathbb R^2;\mathbb C)$ must satisfy three conditions.

First, on its support, each coordinate $x+b_jt$ lies in $I$, at distance at least $\delta_0r$ from the endpoints:

Second, cancellation holds along each coordinate direction: for every $v\in\mathbb R$ and $0\le j\le3$,

Third, for nonnegative integers $a,e$, with $n=a+e$,

The corresponding local quadrilinear form is

The factor $r^{-2}$ matches the scale of the two-dimensional integral. When $rH_I$ occurs below, the prefactor of the kernel is $r^{-1}$. This normalization is essential to recovering the exact difference between scales.

1.3 Amplitude and variation within a block

For a depth block $B=\{u,\ldots,v\}$ and a sequence of functions $(z_s)$, define the pointwise quantity

This controls both amplitude and total variation within the block. Jumps between different adjacent blocks are not counted.

C Theorem 2.2 (pp.3–4, equation (2.4)) makes the following claim. For each position $j=0,1,2,3$, give measurable functions $z_{j,s}:J\to\mathbb C$ at every depth, and partition $\{0,\ldots,d\}$ into $W_j\ge1$ deterministic, nonempty consecutive blocks. Suppose every block satisfies

Extend all functions by zero outside $J$. Then, for any subset $\mathcal I$ of the intervals in the tree,

The constant is independent of the translation of the dyadic grid, the tree, depth, subset, specific kernels, and the four partitions and their block counts. Functions must be specified at every depth, even if no interval at that depth enters the sum.

Estimate (T) bounds the sum of the absolute values of the individual terms, rather than merely the absolute value of their sum. The application below needs only position zero to vary with depth; the other three positions remain fixed. The theorem is stated more generally.

2. Representing a smooth scale difference by a four-direction cancellation kernel

Source: C Lemma 3.2, p.6. Fix an averaging weight $\varphi$ as defined in Unit 1, and write

2.1 Why four vanishing moments are needed

For $j=0,1,2,3$, a change of variables yields

For $j=0$, the factor $2^j-1$ itself vanishes. The other three cases use the vanishing moments of $\varphi$.

Choose $B\ge1$ such that $\operatorname{supp}\psi\subseteq[-B,B]$, and define

Differentiating four times gives $g^{(4)}=\psi$. If $t<-B$, the integral is zero. If $t>B$, expand $(t-v)^3$ as a cubic polynomial in $v$; all four moments vanish, so the integral is again zero. Thus $g\in C_c^\infty(\mathbb R)$, with support in $[-B,B]$.

Derivatives of order less than four are bounded directly by integrals over a finite interval; the higher derivatives are derivatives of $\psi$. Hence $g$ still satisfies the derivative growth condition from Unit 1, with a possibly larger constant.

2.2 The four differential operators

Take the nonnegative mass-one function $\rho$ constructed in Unit 1, and set

Then $\int u=1$ and $\operatorname{supp}u\subseteq[-1/8,1/8]$. Choose a power of two $D\ge1$ such that

Define

These constant-coefficient differential operators commute. For an interval of length $r=D2^k$, with midpoint $m_I$, define the kernel on $I$ by

This is the kernel to which (T) will be applied.

2.3 Support and distance from the boundary

On the support of $W$, we have $|z|\le1/8$ and $|\sigma|\le B/D$, so

After scaling to $I$, this becomes $|x+b_jt-m_I|<3r/8$. Since each endpoint is at distance $r/2$ from the midpoint, every coordinate remains more than $r/8$ from the endpoints. We may therefore take $\delta_0=1/8$.

Each derivative contributes $r^{-1}$ by the chain rule. Once $D$, the slopes, and the derivative constants for $F$ are fixed, the product rule yields the derivative bound of §1.2, for some $C_0$ depending only on $\varphi$ and the fixed slopes.

2.4 Cancellation along the four directions

For fixed $j$, set

The chain rule gives the exact identity

The kernel is therefore $r$ times the derivative on the right. This factor must be retained. The function being differentiated is compactly supported in $t$, so integration leaves no boundary terms. This proves the four line-cancellation conditions.

2.5 Integration in the spatial coordinate recovers the exact scale difference

Expand $W=\prod_j(\partial_\sigma-b_j\partial_z)F$. Every term containing $\partial_z$ vanishes on integration in $z$, leaving only $\partial_\sigma^4F$. Since $\int u=1$ and $g^{(4)}=\psi$,

Consequently,

This is C (3.4). The role of the first three vanishing moments is now explicit: they ensure that the fourth antiderivative $g$ remains compactly supported. This allows the same kernel both to recover the scale difference and to cancel in all four directions.

3. From dyadic intervals to the real line

Source: C Lemma 3.3, p.7, equations (3.5)–(3.6).

3.1 The real-line inequality

Fix $0\le k_-\le k_+$, and partition the integer scales $\{k_-,\ldots,k_+\}$ into $w$ deterministic, nonempty consecutive blocks. Let the bounded Borel functions $h_1,h_2,h_3,h_{0,k}$ all be supported in a common interval of length $L$, and suppose

for each scale block $B$. We seek to deduce from (T) that

3.2 The total root length and the boundary loss

Set $r_k=D2^k$ and $R=r_{k_+}$. For each $a\in[0,R)$, use the translated dyadic grid

Only root intervals of length $R$ that meet the common support interval can contribute. Beyond that interval, they extend by at most one length $R$ at each end. Their total length is therefore at most $L+2R$.

Restrict all functions to each root interval. All four coordinates of a local kernel lie in its own interval $I$, and $I$ lies in the root, so this restriction leaves the local form unchanged.

Within a rooted tree, scale $k$ corresponds to depth $s=k_+-k$. Reversing the scale order changes neither the maximum amplitude within a block nor the sum of absolute adjacent differences. Position zero therefore satisfies the block conditions of (T). The other positions are fixed, so take $W_j=1$.

Apply (T) on each root, sum, and use the triangle inequality. This yields

3.3 Averaging over translations removes the interval partition

Fix the scale $r=r_k$. The midpoints of intervals of length $r$ are $a+(m+1/2)r$, so

This sum is periodic in $a$, with period $r$. Since $R/r$ is an integer, averaging over $[0,R)$ reduces to averaging over one period. A change of variables for each $m$ then gives

For the second equality, the change-of-variable intervals indexed by $m$ cover the real line exactly, with overlaps only at null boundaries. The third equality is (K).

Average (D) over $a\in[0,R)$. The local kernels are replaced by $\Delta_k$, proving (R2). There are only finitely many scales, bounded supports, and uniformly finitely many relevant intervals. All interchanges of sums and integrals are therefore justified.

4. From real-line functions to integer sequences

Source: C Lemma 3.4, pp.7–8, equations (3.7)–(3.8).

4.1 Why the whole integer configuration is needed

The coprimality condition allows integers $e_1,e_2,e_3$ such that $\sum_{j=1}^{3}e_jb_j=1$. Set $e_0=-\sum_{j=1}^{3}e_j$. Then

Thus, for $y_j=x+b_jt$, we have $t=\sum_je_jy_j$. This is the integer reconstruction identity in C Lemma 2.3.

Coprimality of the whole configuration does not imply that two slopes differ by $1$. For example, $(0,6,10,15)$ satisfies the coprimality condition, but no pair has absolute difference $1$. The reconstruction below uses (E), without dividing by any pairwise slope difference.

4.2 Small intervals around integer points

Now let $h_j,h_{0,k}$ be integer sequences supported in a common interval containing $L$ integers, with the same amplitude and block-variation conditions as in the previous section. Write

Choose $0<\rho<1/4$ such that

Extend each value $h(m)$ constantly over $[m-\rho,m+\rho)$, and set the function to zero elsewhere on the real line. These intervals are disjoint, so the amplitude and pointwise variation conditions are preserved. The real supports lie in an interval of length at most $L-1+2\rho\le L$.

4.3 Why no extra integer configurations appear

Suppose the four real coordinates $x+b_jt$ fall in the small intervals centered at integers $p_j$. Disregarding a null set of boundary points, set

By (E), we have $\eta=\sum_je_j\delta_j$, hence $|\eta|\le\rho E_b$ and $|\xi|=|\delta_0|<\rho$. For each $j$,

The left side is the absolute value of an integer, so it must be zero. Thus $p_j=m+b_jn$, and every contributing real configuration comes from a unique integer configuration.

Conversely, for a given integer pair $(m,n)$, the corresponding real configurations are parameterized exactly by

up to negligible boundaries. The set $P_\rho$ contains a neighborhood of the origin and is bounded, as follows from the constraints for any two distinct linear forms. Its area $A_\rho$ is therefore finite and positive. The regions associated with different integer pairs $(m,n)$ are disjoint apart from boundaries.

4.4 The discrete–continuous kernel error is summable

Write $I_k$ for the real-line integral after this embedding, and define the integer form by

The partition above gives the exact identity

Since $\Delta_k(t)=2^{-k}\psi(2^{-k}t)$,

It follows that $|\Delta_k(n+\eta)-\Delta_k(n)|\le\rho E_b\|\psi'\|_\infty2^{-2k}$. A nonzero difference requires $|n|\le B2^k+\rho E_b$, so the number of relevant integers $n$ is $O_{\varphi,b}(2^k)$, using $k\ge0$. The integer $m$ has at most $L$ possible values, and the product of the coefficients has absolute value at most $1$. Therefore

Summing over $k$, we use $\sum_{k\ge0}2^{-k}=2$. Combine this error bound with the real-line estimate and divide by $A_\rho>0$ to obtain

This is C (3.7). The constant is independent of the scale interval, the partition, and the chosen sequences.

5. Why linearizing the maximum produces scale-varying functions

Source: the first half of the proof of C Proposition 3.5, p.9, from the least maximizing index to (3.12).

5.1 Smooth averages for integer sequences

Let $h_1,h_2,h_3$ be integer sequences bounded in absolute value by $1$, supported on $L$ integers in a common interval $K$. Define

Fix endpoints $0\le k_1<\cdots<k_{w+1}$ and a subset $E\subseteq K$. For each $m\in E$, let $\ell_i(m)$ be the least integer at which

attains its maximum over $k_i\le\ell\le k_{i+1}$. The range is finite, so this specifies the choice completely. Set

Then $|\alpha_i|\le1$ and $\alpha_i d_i=|d_i|$. The complex conjugate is essential: choosing $d_i/|d_i|$ would not in general make the product a nonnegative real number.

5.2 Prefix truncation and variation within a block

On the scale block $k_i\le k<k_{i+1}$, define

For each fixed $m$, this function is constant on a prefix of the block and zero after the stopping position. Its amplitude is at most $1/(4\sqrt w)$, and it makes at most one jump within the block, again of size at most $1/(4\sqrt w)$. Consequently,

If $\ell_i(m)=k_i$, the prefix is empty; if $\ell_i(m)=k_{i+1}$, there is no jump within the block. Both endpoint cases satisfy the same bound. Jumps between different blocks need not be controlled, since the definition in (T) counts only within-block variation.

The stopping index depends on $m$, but the block endpoints $k_i$ remain fixed. This is precisely what (Z) permits. A local estimate for fixed functions would not directly handle these scale-varying test functions.

5.3 Summing the difference kernels recovers the exact oscillation

Since $\Delta_k=\varphi_{2^{k+1}}-\varphi_{2^k}$, we have

Thus the sum inside the absolute value on the left of (Z) equals

The right side is nonnegative, so the outer absolute value can be removed. The largest kernel scale used in (Z) is $k_{w+1}-1$. Enlarging the boundary term to $D2^{k_{w+1}}$, we obtain

This is C (3.12). The factor $\sqrt w$ comes from the normalization of the test function, rather than from a separate probabilistic estimate for the maximum.

6. Finite-orbit transference on general probability spaces

Source: the second half of the proof of C Proposition 3.5, p.9, from (3.12) to the end.

6.1 Fix the endpoints, then choose the truncation margin

Let $S$ be an invertible, bimeasurable, probability-preserving transformation on an arbitrary probability space, and let $|f_j|\le1$. Fix endpoints $0\le k_1<\cdots<k_{w+1}$. Choose $B_\varphi\ge1$ such that $\operatorname{supp}\varphi\subseteq[-B_\varphi,B_\varphi]$, and then fix an integer

The margin $Q$ depends on these endpoints, but remains fixed as the orbit length $M$ tends to infinity.

For each $M\ge1$ and $x\in X$, define the finite sequences

The common support contains $M+2Q$ integers. For $1\le m\le M$ and every $k_1\le k\le k_{w+1}$, if $\varphi_{2^k}(n)\ne0$, then

Every $m+b_jn$ therefore lies in the untruncated interval. The sequence averages satisfy the exact identity

This identity holds at all intermediate scales between the endpoints, not just at the endpoints themselves.

6.2 Integration and measure preservation

In (S), take $L=M+2Q$ and $E=\{1,\ldots,M\}$, then integrate over $x$. All averages and maxima involve finitely many measurable functions, so the integrals are well defined.

For each $m$, invariance of the measure implies

Hence

Divide by $M$ and let $M\to\infty$. The endpoints and $Q$ are already fixed, so the boundary terms vanish. We obtain

This is precisely Proposition 3.5 as used in Unit 1, with a constant independent of the system, functions, and endpoints. Although finite-orbit truncation introduces an endpoint-dependent margin, it disappears when $M\to\infty$.

6.3 Why neither standardness nor mixing is required

The argument only samples measurable functions along finite orbit segments, applies an integer-sequence inequality, and integrates using measure preservation. It uses no conditional distributions, ergodic decomposition, or topological model, so the probability space need not be standard. No decay of correlations appears, so mixing is not required either.

Invertibility gives meaning to the negative times in the integer sequences. If the functions are initially specified only as $L^\infty$ equivalence classes, choose bounded representatives and remove the null set obtained from all integer translates of their exceptional sets, as in Unit 1.

7. Constants and the order of limits

After fixing the slopes and one averaging weight, first obtain $D,C_0$ from Lemma 3.2, then determine $\rho,A_\rho$ from the integer reconstruction coefficients. None of these quantities depends on the later scale endpoints or input functions.

For each specified sequence of endpoints, construct the scale-varying test functions and choose the orbit-truncation margin $Q$. Only then let the orbit length $M$ tend to infinity. The constant in (T) must be uniform in depth and block counts from the outset; otherwise the argument cannot yield (O) from Unit 1.

Together, the first two units establish a conditional deduction: the local absolute-sum estimate in C Theorem 2.2 implies the triple pointwise conclusion for mixing systems in C Theorem 1.1. The proof of Theorem 2.2 itself remains to be examined, including the depth-varying local estimates in D and the precise analytic propositions invoked from H.

Guide to the original text

• C p.3, Definition 2.1 and equations (2.1)–(2.3): admissible kernels and the local form.
• C pp.3–4, Theorem 2.2 and equation (2.4): the depth-varying local absolute sum.
• C p.4, Lemma 2.3 and equation (2.5): reconstruction from the full integer configuration.
• C p.6, Lemma 3.2 and equations (3.3)–(3.4): scale differences and four-direction cancellation kernels.
• C p.7, Lemma 3.3 and equations (3.5)–(3.6): averaging over translated dyadic grids.
• C pp.7–8, Lemma 3.4 and equations (3.7)–(3.8): embedding integer sequences and summing the errors.
• C p.9, the proof of Proposition 3.5 and equation (3.12): complex phases, prefix truncation, and finite-orbit transference.