Pointwise convergence of fourfold ergodic averages for mixing transformations

1 Introduction

Let (X,\mathcal F,\mu,T) be an invertible probability-preserving system. For bounded measurable functions f_1,\ldots,f_d, the multiple ergodic averages are

The general pointwise question asks whether these averages converge almost everywhere for every finite d on every invertible probability-preserving system. Here we treat four functions under the mixing hypothesis. We say that T is mixing if

for every A,B\in\mathcal F. The functions are fixed throughout each convergence assertion. No rate in (1.1) is assumed.

Context and inputs

Birkhoff’s pointwise ergodic theorem treats a single observation along an orbit [3]. Furstenberg’s ergodic proof of Szemerédi’s theorem made products at arithmetic-progression times a central object of study [9]. For weakly mixing transformations, these averages converge in L^2 to the product of the integrals. For arbitrary transformations, Host and Kra established norm convergence of every finite consecutive length using nilpotent characteristic factors [11, Theorem 1.1]; Ziegler independently developed universal characteristic factors and another proof [25]. Norm convergence alone does not control the exceptional lengths at an individual point.

Bourgain proved pointwise convergence for two powers of one transformation [4]. Higher-order pointwise results include K systems [6, Theorem 4.2], weakly mixing systems with suitable spectral restrictions [1], and weakly mixing systems whose pairwise-independent finite self-joinings are all product measures [10]. Huang, Shao and Ye proved pointwise convergence of every finite consecutive length on ergodic measure-distal systems [12, Theorem C], extending the distal investigation of Lesigne [19]. Recent pointwise theorems for polynomial iterates include the bilinear result of Krause, Mirek and Tao [17] and the distinct-degree multilinear result of Kosz, Mirek, Peluse, Wan and Wright [16]. Their degree hypotheses do not cover the four linear iterates considered here. Kuca’s survey [18] gives a broader account of the norm and pointwise questions.

Theorem 1.1 uses three companion results. The multiple-mixing theorem of [22, Theorem 1.1] supplies mixing of every finite order from (1.1). The analytic construction of [21], based on the local estimates of [20], supplies an oscillation estimate for threefold averages. Its oscillation assertion holds on arbitrary invertible systems; mixing is used separately to identify limits. We need the threefold estimate at each three-element subset of \{1,2,3,4\}. Appendix A proves the required coefficient adaptation and the resulting pointwise convergence lemma, including its nonergodic formulation. The local analytic theorems themselves are cited with their hypotheses and locations. Thus the contribution here is the fourfold deduction from these specified inputs.

The structural part uses the Furstenberg–Zimmer theory of relatively compact and relatively weakly mixing extensions. We record the nonergodic and power-stability statements at their point of use in Section 2. The construction also uses extensions on which certain conditional expectations cannot improve. Such energy maximization belongs to the same general method as sated extensions [2]; the data being saturated here come from selected four-coordinate laws.

The difficulty caused by selected lengths

Suppose that the conclusion fails. After subtracting means, there are four centred real functions and a positive-measure set of points at which their averages remain positively separated from zero along arbitrarily late lengths. We choose one such length N_a(x) after scale 2^a. Averaging the configurations

and passing to a limit preserves a nonzero correlation. A constant-roof suspension makes these limit laws invariant under a flow and permits the changes of speed needed later.

Selection is the main analytic obstruction. Conditioning on a length chosen from the initial state can change the joint law of independent auxiliary variables. Ordinary norm convergence, with no rate, does not by itself bound all the resulting errors. We introduce three groups of auxiliary spatial coordinates. Their joint law is a product, and any two groups together are independent of the original coordinate, after conditioning on the suspension phases. A length event may change the full three-group law, but preserves the two-group marginals. This is precisely the amount of independence used in Section 5. For a finite model of the distinction, let U,V,W be independent symmetric signs. Their product is independent of each pair, but conditioning on UVW=1 constrains the triple.

The estimate making this possible is a finite-dimensional packing statement proved in Section 4. If two bounded families of Hilbert-space vectors have small pairwise inner products outside a graph of bounded degree, few of their tensor products can have large projection onto a subspace of slowly growing dimension. A Gaussian comparison and a trace-moment estimate give a quantitative form of this assertion. Applied to the leading covariance subspace, it controls the projected part of each selected average. The complementary part has a square-summable error over the possible lengths. The estimate is independent of the ambient Hilbert-space dimensions and requires no quantitative mixing rate.

From a correlation to a contradiction

Each auxiliary system carries both an original spatial coordinate and the three groups of additional coordinates. We call such a system a channel after specifying its exact independence conditions in Definition 5.1. In its selected four-coordinate law, consider nonzero correlations in which each slot uses either a centred function of the original coordinate or a centred function of the additional coordinates. Bounded functions of the distal parameters are allowed as weights. The initial failure gives a correlation with no additional-coordinate slots.

Choose a channel and a correlation maximizing the number of slots of the second kind. Independence excludes four such slots. An iterated array construction excludes three: products along different axes through a fixed corner are orthogonal, but each has the same nonzero correlation with the function at that corner. Bessel’s inequality rules this out. At most two slots can therefore be of the second kind.

The final construction increases that number. First enlarge the channel until the conditional expectations needed by the argument are stable under further extensions. Then regard one selected four-coordinate law as a new system and select once more. The result is a square of old states. Its distinguished row and column meet at one corner; saturation supplies their exact conditional laws. A reanchoring identity on the distal parameters shows that the row and column correlation cannot vanish. Regrouping the other coordinates therefore replaces one original-coordinate slot by an additional-coordinate slot, contradicting maximality.

Sections 2 and 3 establish the structural inputs and selected laws. The packing estimate and channel preservation are proved in Sections 4 and 5. Section 6 carries out the maximality argument and completes the proof. The coefficient-dependent analytic work is collected in Appendix A.

2 Convergence inputs and relative structure

The auxiliary systems in the proof will be extensions and joinings of a suspension of the original transformation. They need not be ergodic. We therefore formulate the structural facts in the nonergodic setting, and distinguish the pointwise input for three functions from the norm estimate available for any finite number of functions.

2.1 Standard systems and the pointwise inputs

Put I=\{1,2,3,4\}. It suffices to prove Theorem 1.1 on a standard probability space. Indeed, for a given tuple of bounded functions, the map recording all their integer translates takes values in a countable product of compact scalar discs. The pushforward measure and the shift give a standard factor, the functions descend to its coordinate functions, and mixing passes to this factor. Almost-sure convergence there pulls back to the original space. We henceforth write (X,\mu,T) for this standard mixing system. All factors and conditional expectations are understood modulo null sets; probability spaces may be completed.

We use [22, Theorem 1.1] in the following form. For every fixed finite family g_1,\ldots,g_s\in L^\infty(\mu),

The bounded-function formulation follows from the set formulation by simple-function approximation. All uses of this input involve fixed functions; no rate of mixing is assumed.

The other companion input will be used on systems without mixing. Its coefficient dependence is important, so we isolate its exact scope.

The proof, including the changes of coefficients in the analytic estimates of [21, 20], is given in Appendix A. Neither ergodicity nor mixing is an assumption of Lemma 2.1.

2.2 Distal factors and relatively weakly mixing extensions

Let (Y,m,R) be a standard probability-preserving system, and let B be a factor. We use the conditional-correlation characterization of relative weak mixing: for bounded f,g, with \E[f\mid B]=0,

For F_1,\ldots,F_r\in L^2(m), the associated finitely generated L^\infty(B)-module is

It is invariant if composition with R and with R^{-1} preserves this space. An extension is relatively compact if its L^2 space is generated by these invariant modules. The individual modules need not be closed in L^2; closure is taken after their union is spanned. This is the compact-module characterization in its corrected form [13, Theorem 4.1][14, Theorem 2.5]. A relatively distal extension is a tower of relatively compact extensions and inverse limits. Distality without a specified base means distality over the trivial factor. These definitions include nonergodic systems; in particular an identity action is compact.

The following norm estimate explains why proper subaverages have a particularly simple law over a distal factor. Its proof uses relative weak mixing alone.

In particular, weak mixing of T gives L^2 convergence to the product of the means for all finite distinct nonzero linear powers on X. In a flow, with B preserved by every time, the analogue of (2.4) for deterministic continuous-time averages follows by writing t=n+s, 0\le s<1. Apply the lemma for each fixed s to g_i\circ S_Y^{a_i s}, and integrate the resulting L^2 estimate in s. Conditional expectation commutes with these time shifts. Bounded convergence and the final interval of length less than one complete the passage to continuous time.

3 Selected laws and their distal parameters

A failure of pointwise convergence allows the averaging length to depend on the initial state. We encode this failure in a probability law on four outputs. The selection can change their joint law, but the convergence results of Section 2 will determine every proper marginal, even after all distal parameters are given.

3.1 From failure to a sequence of selectors

Let C=X\times[0,1) have measure m_C=\mu\otimes\dd u and the roof-one suspension flow

Write \xi:C\to X for its spatial coordinate and \upsilon:C\to\R/\Z for its phase. The use of a flow will let us normalize the action at any one of the four outputs.

For the rest of the proof assume failure and fix the functions, selectors and phase test of Proposition 3.1. We use one fixed free ultrafilter \mathcal U on the indices a.

3.2 A functorial law on four outputs

Consider a standard probability-preserving flow (Y,m_Y,S_Y^t) with an equivariant factor \pi_Y:Y\to C. The selector on Y is N_a\circ\pi_Y. For bounded measurable functions G_i on Y, define the selected law by

The existence of a countably additive law with this formula is part of the next proposition.

Let \omega_Y:Y\to W_Y be the maximal distal factor of time one, write \nu_Y=(\omega_Y)_*m_Y, and disintegrate m_Y=\int m_{Y,w}\,\dd\nu_Y(w). For a state (y_i)_{i\in I} of the selected law, put P_i=\omega_Y(y_i) and P=(P_i)_{i\in I}.

For b\in I define the normalized flow

Its projection to slot b is an equivariant extension of Y and therefore of C. This permits the selected-law construction to be iterated with the same selectors and the same ultrafilter.

Finally, the phase test in Proposition 3.1 can be approximated uniformly by finite sums of products of one-slot phase functions. Formula (3.4) therefore applies to its product with the four spatial tests. In particular,

The phase array is part of the distal parameter array, since phase rotation is compact. Equation (3.8) is the nonzero centred correlation from which the argument of Section 6 will start.

4 Packing tensors into a subspace of small dimension

Selecting an averaging length changes the joint distribution of the variables being averaged. The deterministic Hilbert-space estimate in this section will control that change. In the application, two independent groups of variables supply the two L^2 spaces whose tensor product we consider. The estimate is uniform over the testing subspace: a subspace of moderately growing dimension captures a substantial part of only a small fraction of the tensors in an almost orthogonal family.

Throughout this section the Hilbert spaces are real, tensor products are Hilbert tensor products, and logarithms are natural.

The proof separates directions on which the testing space has large variance, then applies a Gaussian comparison to the remaining directions. The comparison is the finite form of the Sudakov–Fernique inequality; see [7, Section 1, Equation (1.2)] and the minoration in [23]. We first record the matrix estimate needed for this application. Its proof is the Gaussian integration-by-parts moment argument underlying matrix Khintchine estimates; compare [24, Lemma 6.1 and Proposition 7.3]. We include the derivation to display the dependence on the covariance trace rather than on the ambient matrix dimensions.

We will use the conclusion also in averaged form. Since every projection norm is at most one, for any fixed accuracy \varepsilon, vectors satisfying the corresponding small-pair hypothesis, and an orthogonal projection P of rank at most d_m,

Here \varepsilon and then \eta are fixed before m increases. The exceptional degree may depend on this chosen \eta. In Section 5, mixing supplies precisely this order of quantifiers; a uniform rate of mixing will not be needed.

5 Channels and preservation of independence

The selected laws of Section 3 can retain correlations that ordinary averaging would destroy. We now specify a class of extensions carrying independent output variables and show that this class survives passage to a selected law. There are three groups of outputs. Every pair of groups is independent of the original input, whereas their full joint law may be correlated with that input. This distinction is what permits selection of lengths and still leaves room for the correlations used in the final argument.

Write a point of the suspension C as (x,u), with spatial coordinate x\in X and phase u\in[0,1). A factor of a flow Y with values in C and speed q>0 means a measure-preserving map D:Y\to C satisfying D(S_Y^t y)=S_C^{qt}D(y) for every t, modulo the usual null sets.

In particular, the original spatial variable is independent of every proper collection of groups, even after the phases are given. The definition does not require its independence from all three groups jointly. The flow C with no outputs is a channel.

Recall that \lambda_Y is the selected law on Y^I, and that \mathsf M_b(Y) equips this law with the flow whose speed in slot i is i/b. Projection to slot b is then an extension of Y.

The new output speeds are iq_\alpha/b, so they remain positive and rational. The content of the proposition is independence. Its proof first treats a proper collection of groups, for which conditioning on the selected length preserves independence from the input. For all three groups, the selected law can change, and tensor packing provides the needed estimate.

The channel and its tests are fixed before the selected limit is taken. The packing estimate and the summation over length classes require no uniform mixing estimates over different channels or over all iteration depths.

6 Activating an additional output role

The channel construction supplies independent spatial outputs even after the averaging length has been selected. We now use that independence to eliminate the nonzero correlation supplied by Proposition 3.1. The argument counts how many of its four factors can be read from channel outputs. After bounding this number and choosing a correlation with the maximal number of output roles, we take an extension that stabilizes certain conditional expectations. This will permit a correlation with one additional output role, contradicting maximality.

Throughout this section the selectors, and hence the assignment Y\mapsto\lambda_Y, are fixed. For a channel Y, write d_Y(y) for its full list of spatial outputs and x_Y(y) for the spatial coordinate of its factor \pi_Y(y)\in C. If the output list has length m, its spatial product law is \mu^{\otimes m}; when m=0 this is the one-point probability space. All functions used below are real and bounded.

6.1 Correlations and an upper bound on output roles

For A\subseteq I, consider a correlation

Here \Phi is an arbitrary bounded measurable function of the complete distal parameter array. Each h_i is centred for its spatial law: \mu^{\otimes m} when i\in A and \mu otherwise. We call the indices in A the output roles. The functions in an output role may depend jointly on the whole output list. Lemma 5.2 implies that every factor in (6.1) is also centred conditionally on its own parameter P_i.

The channel with no outputs is permitted by Definition 5.1. The phase test in Proposition 3.1 is measurable in the distal parameters of the suspension. Thus that proposition gives a nonzero correlation (6.1) with A=\varnothing. Consequently, under the supposition of pointwise failure, there is a largest number r\in\{0,1,2,3,4\} of output roles in any nonzero correlation of this form, over all channels and all such tests.

6.2 Stabilizing conditional expectations under extension

The next extension construction prevents extra information in a later system from changing the conditional laws that we shall use. It is an energy-increment construction of the kind used for sated extensions [2, Theorem 3.11 and Lemma 3.12]; we give the argument for the selected-law assignment needed here.

On (Y^I,\lambda_Y) introduce the five sigma-algebras

If \rho:Y'\to Y is an extension over C, functoriality gives the factor map \rho^I:(Y'^I,\lambda_{Y'})\to(Y^I,\lambda_Y). The pullback of each \mathcal D_l(Y) is contained in \mathcal D_l(Y'), since the old distal factor is contained in the new one.

We call a system satisfying Proposition 6.2 saturated. A nonzero correlation persists under the passage to a saturated extension: its functions pull back, and its old parameter test remains measurable in the enlarged distal parameters.

6.3 A channel carrying the three other factors

Choose a nonzero correlation with the maximal number r of output roles, and replace its channel by a saturated extension Y. Retain the notation A,\Phi,h_i for its tests. Proposition 6.1 gives r\le2, so choose b\in I\setminus A. Set

Its state law is \lambda_Y. We give Z a new channel structure adapted to this particular correlation. Keep the full old output list from slot b, and for every j\ne b add the list

where in the second case the actual flow factor added is the copy of C in that slot. Index the new three groups by j\in I\setminus\{b\}. Put Q_j in group j and assign the three old groups from slot b bijectively to these new groups. All speeds are positive rationals.

Write

It is a function of the three added lists Q_j in a single Z-state. With y=y_b, define

The original nonzero correlation equals \E[\Phi(P)h_b(x_Y(y_b))g(P,y_b)], so g is nonzero in L^2. The two conditional expectations in (6.6) use the same state law, but different parameter information: e contains the old array P and may contain more.

6.4 The square and its conditional laws

The selected law \lambda_Z is a law on a square of Y-states. Write \mathbf z_i=(y_{ij})_{j\in I} for row i, a Z-state, and set

Each E_i contains the complete old parameter row (P_{ij})_j, by Lemma 3.3. In column b put

Functoriality for Z\to Y says that this column has law \lambda_Y. Row b has the same law because a single Z-state has law \lambda_Y. Our aim is to show that

The row factor q(\mathbf z_b) is read from the added outputs, so it is the candidate for the additional output role at b. To establish nonvanishing, saturation will first identify the column law given E and the conditional expectation of the row factor given that column. Figure 1 distinguishes these two uses of the old selected law.

We can now compute the correlation between the two highlighted lines. First condition on the column in Lemma 6.4, then condition within its old law on its corner state. The result is

The right side is not manifestly positive. The next lemma supplies the parameter independence that will show it cannot vanish identically.

6.5 Nonvanishing and the additional role

We record the elementary Hilbert-space fact used to finish the argument. Its separability hypothesis is automatic for the probability fibres occurring here.

A Pointwise convergence for the proper subsets of four times

We prove Lemma 2.1. The systems to which that lemma is applied in Section 3 are extensions of a suspension and need not be mixing or ergodic. Accordingly, the input needed here is a pointwise convergence theorem on arbitrary invertible probability preserving systems. Its analytic basis is the oscillation estimate in [21, Proposition 2.2], denoted by P in this appendix. We also write H for [20].

P states this oscillation estimate for the times n,2n,3n before using mixing to identify its limit. We need the three additional triples contained in \{1,2,3,4\}. We obtain them by modifying the local four-linear construction in H and its extension to inputs varying with depth in P. This is a use of those analytic proofs, including their intermediate estimates; the boundedness theorem for the trilinear Hilbert transform alone would not supply the required oscillation estimate. We specify the imported statements first, then verify the coefficient changes, and finally give the passage to arbitrary systems and all integer averaging lengths.

A.1 The local estimates used from H and P

For this appendix, a role set is a set S of four distinct integer coefficients. Initially S=\{0\}\cup J, where J\subset\{1,2,3,4\} has three elements. If K is a dyadic interval of length r, put

Fix c_0>0 and C_0\ge1. The admissible kernels are the smooth, compactly supported functions w_K for which every x+jt on the support lies in K at distance at least c_0r from its boundary,

Kernels may be chosen independently at the different intervals.

H’s Uniform local estimate [20, Theorem 2.2] asserts, for S=\{0,1,2,3\} and a suitable q\in(2,3), that

for every finite family of dyadic intervals and compactly supported scalar L^q inputs. Here M is the Hardy–Littlewood maximal operator. Its proof, in H, Sections 3–10, supplies the quantitative chart, projection, separation, counting, and side estimates used in P.

The extension needed for oscillation is P’s Local estimate with variation in depth, Theorem 3.2. To state it, let K_* be a root interval and let s(K)=\log_2(|K_*|/|K|) be the depth of a descendant. For a finite integer block B=\{a,\ldots,b\} and a scalar sequence (z_s), write

For the original role set \{0,1,2,3\}, P proves the assertion below. We establish it for the four required role sets.

The proof of Proposition A.1 is the proof-by-modification below. The analytic inputs are H’s local estimate and its proofs of the intermediate estimates, and P’s depth-local construction in Sections 4–7. Proposition A.1 is the sole analytic input to the local-to-orbit argument in Subsection A.6; Subsection A.7 then proves Lemma 2.1. The intervening subsections establish the proposition by checking, in order, the coordinate and quadratic identities, the integer-lift and counting arithmetic, and their use in P’s depth scheduling. We retain their definitions of quadratic chart atoms, inherited projection families, curvature tags, and finite progression tests. The following computations specify their dependence on the four role coefficients. In particular, the proposition is not an assertion of uniform continuity of a Hilbert-transform bound in its slopes.

A.2 Coordinate changes and the quadratic cancellation vector

Every set S above contains an adjacent pair. Translate all role labels by the same integer so that one such pair is 0,1. This is the change of variables x'=x+mt in (A.1), with Jacobian one. The four spatial coordinates themselves are unchanged. Consequently the support and cancellation conditions remain the same, and the derivative bounds retain their form after increasing the fixed constant C_0. During the local argument we use these translated labels. Their pairwise differences have absolute value at most four, and their diameter is either three or four. Thus

The vector (-1,3,-3,1) in H is replaced by a primitive integer vector (c_j)_{j\in S} proportional to

The overall sign is immaterial. Before translation, convenient choices are

Lagrange interpolation of the polynomials 1,u,u^2 gives

These identities are preserved by translation of the role labels. Every c_j is nonzero. All coefficients and denominators arising from this finite list of coordinate systems are bounded absolute constants.

We distinguish numerical role labels from enumerations of four objects. A coefficient such as j, j-i, or c_j is changed according to S. An ordering of four input sizes, four leaves, or four positions in a multilinear expansion still enumerates four objects. This convention leaves those parts of H’s proofs unchanged.

A.3 Chart separation, detection, and the scale estimates

The local absolute estimate, H, Lemma 2.1, uses the map (x,t)\mapsto(x+it,x+jt), whose determinant is j-i. Its endpoint bounds and interpolation therefore hold for the new roles. In particular the initial bound for the finite-depth constant in (A.3), and H’s stopping reduction to normalized local sizes, remain valid. The coefficient and projection bounds, compression and height-bin estimates, H, Lemmas 3.2–3.5, are statements about finite families of vectors or real frequencies and do not specify a progression coefficient.

We next check the part of H, Section 4, that converts two chart atoms in distinct roles into separated factors. A chart phase has the form

where \theta is a vector of real frequencies, C is the rational chart matrix, and d is a bounded lift of \theta y modulo the integer lattice. The source definition also includes a smooth amplitude on the lift. Its one-variable Weyl estimates, Fourier expansion, and uniform tail estimates are unchanged.

For H, Lemma 4.5 (Progression separation), fix source roles i,k and spectator roles a,b. Set \alpha_i=1, \alpha_k=0, and solve

for \alpha_a,\alpha_b. This is possible because a\ne b. Writing y_j=x+jt, insert the same lift partitions as in H at \theta y_a, \theta y_b, and \theta t. If their lifts are d_a,d_b,d, then

is an integer vector. For instance, if d_j=\theta y_j-m_j and d=\theta t-m, the identity y_j=y_a+(j-a)t gives k_j=-m_j+m_a+(j-a)m. Thus the equality cutoffs in H still test integer carries exactly. Their support bounds change only by a fixed factor. With A_2=\sum_jj^2\alpha_j, direct expansion gives

The last sum is linear in x,t and hence separates in y_a,y_b. The smooth amplitude on the bounded lift box has the same Gevrey bounds up to fixed constants. Fourier expansion therefore gives H’s absolutely convergent separation into two spectator atoms and one atom in t, with the same coefficient-tail exponents and quasipolynomial complexity bounds. H, Lemma 4.6 (Single-scale separation), then uses t=(y_b-y_a)/(b-a); every added rational height is bounded by an absolute constant.

For the generalized von Neumann bound, H, Lemma 4.11, let i be the role whose input is controlled by its U^3 norm. Apply Cauchy–Schwarz in the directions (-j,1) for the other three roles, beginning with the role containing the L^2 input. Each direction holds x+jt fixed and changes x+it by (i-j)h. Multiplication by the nonzero integer i-j preserves Haar measure on a circle. The three independent circle parameters therefore produce the same cube integral as in the source proof. Character weights split using the marked pair x=y_0, t=y_1-y_0. Enlarging the fixed periodic boxes accommodates all translated role sets. This verifies the coefficient-dependent part of the generalized von Neumann bound. In H, Lemma 4.12 (Local detection), the potentially unbounded L^2 residual occupies a different role. After clipping the opposing inputs, the source proof decomposes two of them using its continuous inverse theorem and bounds the uniform errors by Lemma 4.11. The separated chart moments then detect the residual. Those separations were verified above; the clipping, inverse theorem, and mixed-norm exponents are unchanged.

H, Lemma 5.1 (A scale sum for one separated term), uses the spectator coordinates (y_a,y_b). Holding one fixed, the integral of the kernel over the other is zero by (A.2), with a constant Jacobian factor |b-a|. The double primitive of that kernel consequently has the same compact support and smooth bounds. Its tensor expansion into mean-zero bumps, the Bessel estimates for those bumps, and the frequency-height bounds apply with the new pair. The only additional demodulations have frequencies \pm\gamma/(b-a); their rational multipliers have bounded height, although \gamma can be arbitrary real. This also verifies the coefficient changes in the finite predictor list and the delayed predictor replacement. The scale bands, accuracy gaps, and projection-energy bounds in the remaining base construction depend on these estimates and the four-slot multilinearity, rather than the values of the labels. There is also a multivariable use of the chart estimates in H, Lemma 5.7 (Few active scales for a fixed triple). Squaring the transpose at the remaining role \ell gives six atoms with arguments h+(j-\ell)t and h+(j-\ell)t'. These are still fixed integer linear forms in three variables, so the three-variable Weyl alternative and chart expansion of H, Lemmas 4.3–4.4, apply with the same complexity bounds. Its final kernel cancellation is the marginal at anchor \ell in (A.2).

A.4 Curvature tags, short blocks, and integer heights

H’s localization in Section 6 groups chart atoms by their normalized quadratic phase and horizontal chart data, called their curvature tags. Normalize the parameters (\lambda,C) in role j by dividing both by c_j, and retain the horizontal vector \theta. The coefficient calculation in H, Lemma 6.1 (Separation of different tags), can then be checked directly. For source roles i,j and spectator roles a,b, put

The barycentric formula gives

Indeed the factors (i-a)(i-b) cancel from the product defining c_i, leaving a nonzero multiple of (i-j)^{-1}; interchanging i,j changes its sign. The return calculation in that lemma therefore has the same nonzero prefactor and the same separation conclusion. Its period is enlarged to clear these fixed denominators.

The residue-template calculation in H, Lemma 6.6 (A uniform same-tag test), uses the marked adjacent pair. Its lifts satisfy

Substitution of (A.10) and (A.7) cancels the common quadratic terms and leaves only linear phases from the carries. Their Fourier expansion has the same residue-template bound after a fixed enlargement of the lift box. It introduces no new curvature tag. Thus the source localization proof receives the same tag-separation and transfer estimates. Its graph distances, color histories, masks, and projection bounds are then used with those estimates and their original parameter order.

H, Lemma 7.2 (Short-block comparison), uses two sampling windows to compare a finite lattice form with a form on a connected torus. Place those windows on roles 0,1. The lattice and lifted torus coordinates in the other roles are respectively

Both formulas have integer coefficients. A fixed torus of length 64 and lift boxes in [-12,12] accommodate the present role sets; the source support parameter is decreased by a fixed factor if necessary. Maps from a pair of variables to two distinct role coordinates are injective on their lattice images and surjective on connected tori. These are the pair-coordinate properties needed in the comparison.

The quadratic randomization in role j becomes e(\gamma m^2/c_j), where e(v)=\exp(2\pi i v) and \gamma is uniform modulo a common multiple of all |c_j|; the value 24 works for (A.6). On the Fourier support of the four-linear form,

The two vectors c=(c_j) and (jc_j) are independent and span this two-dimensional nullspace. Hence m_j=c_j(v+jv') for some real v,v', and

by (A.7). The randomization preserves the form. For each single role, averaging its fourth moment in the spatial variable and in \gamma imposes equality of a pair’s sum and sum of squares; the same two pairings as in H remain. Thus its fourth-moment and clipping estimates, including their powers of q-2, are unchanged.

The return from lattice points to small intervals also holds with these coordinates. If four real role coordinates are within e of integers, the marked adjacent pair reconstructs an integer progression. The discrepancy at any other role is an integer of magnitude less than 8e. Choosing e<1/100 forces it to vanish. The admissible real offsets have positive area, computed explicitly below in (A.16). All support and overlap losses in the short-block comparison are therefore fixed constants.

The remaining arithmetic change is the height base. In H, Sections 8–9, replace L_h=6^h by L_h=12^h, and replace the factor 36 in the finite progression tests by 144. Anchor changes still have integer coefficients. The common progression multiplier in H, Section 8.3, uses exactly (A.10) and (A.7), so its smoothness and Fourier truncation are unchanged in type. For the divisibility step, a coordinate in the free lattice quotient in H, Lemma 8.3 (Height stabilization), is an integral rational multiple of 12^{R'}. Here the quotient is the integer coordinate lattice modulo its intersection with the source’s small rational subspace, R' is the height gap, and C_* bounds the denominators before multiplication by 12^{R'}. The intersection is saturated, so this quotient is free abelian. Writing \Delta for the required drop in height, its 2- and 3-adic valuations are at least

Thus R'>\log_2C_*+2\Delta ensures divisibility by 12^\Delta. No assumption that the initial denominator has only primes 2,3 is needed: the resulting coordinate is already integral. The reserves and the later growth constant A_0, used in the source bounds E_h=A_0^{h+1}, are chosen using the new base.

A.5 Progression counting and the depth-varying extension

We finish the coefficient check at the finite counting step because it requires more than the identities for quadratic phases. In H, Section 9, eliminating a role \ell' gives the three coefficients

They are nonzero and have absolute value at most four. The horizontal frequency tests use sums of role frequencies, while their anchored tests use the weights i-\ell'. Their kernel cancellations are precisely (A.2). Accordingly the source’s Fourier expansions, coefficient tails, and annular tests keep their forms after these replacements.

In H, Lemma 9.1 (Sparse mass of separated progression tests), the shift data consist of a vector \zeta\in\R^d and integer vectors n with \|n\|_\infty\le B. Replace the shifts 36n\cdot\zeta in its annular relations by 144n\cdot\zeta, and the neighborhood shifts 6n\cdot\zeta by 12n\cdot\zeta. The parameter N_0 is the coefficient reserve for the latter neighborhoods. The proof uses a fine common grid; choose its step on a multiple clearing the integers 1,2,3,4.

For a four-step path, alternately add its four annular relations. The intermediate label in role i and the two labels in a second role cancel, leaving \alpha_i(a-a') and an expression in the four color labels. Dividing by \alpha_i constructs the source’s difference cover. The required lattice compatibility is

Both assertions hold because |\alpha_i| divides 12. The integer vector obtained by adding four original shifts has norm at most 4B. In units of the new neighborhood shift 12, division by \alpha_i therefore gives a vector of norm at most 4B\cdot12/|\alpha_i|\le48B. The reserve N_0\ge64B accommodates it. The scalar error is at most four times one relation’s error after division by a nonzero integer.

For the slab cover, write the first two coordinates as their box centers plus shifts 12n_i\cdot\zeta and scalar interval errors. Substitute them in the three-role relation and solve for the third coordinate. The resulting shift coefficients are 12\alpha_1/\alpha_3, 12\alpha_2/\alpha_3, and 144/\alpha_3. By (A.11) they are integers, and the resulting shift vector has norm at most C(d+1)^2B as in H. Choose the separation reserve and the excluded pure-shift interval constants larger by the fixed factors arising from these substitutions. The covering iterations use the same orders of sumsets, the graph count uses the same four roles, and the block and scale counts use the same covers. Their exponents are consequently unchanged. In particular the source factor L^{29}\delta^{1/2080} remains valid, where L bounds the number of scale occurrences of a fixed pair of labels and \delta is the sparse tuple-density threshold. In the application, let \xi be the vector of frequencies obtained from the height construction and take \zeta=\xi/144, so the available heights start at 12^{\Delta-2} instead of 6^{\Delta-2}. The preceding height reserves supply the required shift sizes. Pair determination in the application divides only by role differences, and the short-block windows were checked above. This verifies the coefficient-dependent hypotheses of the counting estimate and its application, not merely its formal conclusion.

We can now apply the remaining operator and summation arguments of H. The main projection families, Fourier modes, and counting tests have the same bounds in powers of logarithms of their budgets. The coordinate changes have inserted only fixed rational heights, fixed derivative losses, and the replacement of one fixed exponential height base by another. None changes a budget exponent as a function of the final q. H’s Section 10 chooses those exponents before taking q\in(2,3) sufficiently close to two; its absorption of the finite-depth constant therefore applies to the modified estimates. This proves (A.3) for the four role sets together with the intermediate estimates used in P.

It remains to carry these inputs through P’s depth-local proof. The Maximal estimate for local catalog rows, The depth-varying row estimate, and Smooth mean-zero rows on depth inputs in P, Section 4, concern Hilbert-space operators supported on nested dyadic cells. Here a catalog consists of finitely many vector-valued column functions, each with a dyadic birth cell; descendants use restrictions of the same functions. A local projection is the orthogonal projection in the cell’s L^2 space onto constant-coefficient combinations of a specified set of available columns. Their hypotheses specify cellwise constant coefficients in a persistent catalog, an ordinary fixed-input Bessel bound, and nondecreasing input-read indices. They contain no role coefficients. These are P, Lemmas 4.3, 4.4, and 4.6. The replacement just proved supplies their same scalar and Bessel inputs.

More concretely, P, Lemma 5.3, Corollary 5.4, and Lemma 5.5, use (y_a,y_b) and the demodulations \pm\gamma/(b-a) already checked. Their cutoff equal to one on [-1/3,1/3] still covers the support by (A.5). Separating the kernel gives exactly mean-zero bump rows, with fixed complete carriers. Clump replacement and predictor rows have the same Bessel bounds because the cancellation and height estimates have just been verified. The support and observation partitions of the rows, and their nondecreasing reads of the depth inputs, are unchanged. In P, Section 6, use the normalized tags from (A.7); the mask, template-transfer, short-block, and counting estimates are then the versions established above. Scheduling the column updates and summing their row bounds introduces no new arithmetic operation on a role label.

For completeness, the final parameter dependence can be seen in the retained estimates of H, Section 10.3, and P, Section 7. Write k for the accuracy logarithm, \mathfrak p for their growth parameter, K_0 for the resulting depth-block length, and G_0 for the strong energy and projection overhead. The term requiring q close to two has the form

Here we have renamed the source parameter P to avoid confusion with the manuscript P. The degrees C,C_5,C_{19} have absolute bounds fixed before q. The short-block estimate feeding this term has the explicit exponent 11(q-2)/(q-1) in H, Lemma 7.2; the coefficient changes above affect its fixed constants, not this exponent. Taking q so close to two that

is therefore permitted in both proofs. Here the accuracy scheme fixes 0<\theta<1 before q, and the positive constant \alpha is chosen sufficiently small after q, as in the source proofs. Their later choice \mathfrak p=\lceil\exp(\alpha k^\theta)\rceil gives G_0=\mathfrak p^{o(1)} and makes the displayed term at most C_q\mathfrak p^{-1/4}. The remaining accuracy exponents and summations retain their source order of choice. P’s depth-local proof consequently gives (A.4).

Translating the role labels back restores S=\{0\}\cup J. P’s depth-local theorem permits variation in any role, and the local constructions allow any role as testing slot. Thus this translation does not require the testing role to remain the marked role 0 during the local proof. This completes the proof of Proposition A.1.

A.6 From local forms to orbit oscillation

An averaging profile is a real compactly supported smooth function \varphi satisfying

for some G\ge1. Profiles may have either sign. Write \varphi_A(t)=A^{-1}\varphi(t/A). These are the profiles of P, Section 2.

A.7 Convergence on arbitrary systems

We now prove Lemma 2.1 from Proposition A.2. The use of quantitative oscillation bounds to prove pointwise convergence belongs to the classical approach developed in [15]. The criterion needed here has the following short proof. A uniformly bounded sequence of measurable functions satisfying (A.13) is Cauchy almost everywhere. Indeed, if its tail oscillation exceeds 4\epsilon on a set E of positive measure, then for every fixed initial endpoint k the supremum of |V_\ell-V_k| over \ell\ge k exceeds 2\epsilon on E. By monotone convergence one can choose a deterministic later endpoint so that the corresponding integral maximum is at least \epsilon\nu(E). Recursively choosing endpoints gives a lower bound W\epsilon\nu(E), contradicting the upper bound C_\varphi\sqrt W for large W. The finite sums defining V_k^\varphi are uniformly bounded because the profile is bounded and compactly supported. Thus every fixed profile gives almost everywhere convergence.

P, Lemma 2.8 (Approximation with three vanishing moments), states that for every c\in[1,2] and every \epsilon>0 there is an averaging profile with

The use of signed profiles is essential to this assertion. One first smooths h_c, preserving integral one. The other three moments can be canceled, without changing the integral, by four broad translated bumps at distance and width comparable to a large parameter A. Their moment matrix for orders 0,1,2,3, after division of row j by A^j, is a fixed invertible Vandermonde-type matrix. The prescribed zeroth correction is zero and the remaining normalized corrections are O(A^{-1}). The correcting coefficients are therefore O(A^{-1}), and their total L^1 cost tends to zero. The bumps retain the derivative bounds in (A.12) with a profile-dependent constant.

Fix a triple J and write a_n(y)=\prod_{j\in J}g_j(R^{jn}y), so |a_n(y)|\le1. For N_k=\lfloor c2^k\rfloor, the Riemann-sum comparison gives, uniformly in y,

To see this, first replace N_k^{-1} by (c2^k)^{-1}, at cost O(2^{-k}), then bound the difference by 2^{-k}\sum_{n\in\Z}|h_c(n2^{-k})-\varphi(n2^{-k})|. This converges to the displayed integral because the integrand is compactly supported and piecewise continuous.

Choose countably many profiles with errors tending to zero for each rational c\in[1,2]. On their common conull set, the existence of all the smooth-profile limits and (A.18) show that the ordinary averages along N_k are Cauchy. The Host–Kra norm convergence theorem [11, Theorem 1.1], stated for arbitrary invertible probability preserving systems, gives an L^2 limit for the averages at all integer lengths: apply it to the four consecutive powers, inserting the constant function one at the omitted role. The bounded pointwise limit along each N_k must equal this L^2 limit almost everywhere, by bounded convergence and uniqueness of norm limits. Thus all rational-c subsequences have the same limit on one conull set. No identification of that limit and no mixing assumption is needed.

Finally, for M\le N and |a_n|\le1,

For any rational mesh of [1,2], every sufficiently large integer N lies within its relative mesh size, up to O(N^{-1}), of some \lfloor c2^k\rfloor from that finite mesh. Its finitely many subsequences have the common pointwise limit. The preceding inequality, followed by meshes of size tending to zero, proves convergence along all positive integers. Homogeneity removes the normalization |g_j|\le1. A proper subset with fewer than three roles is contained in one of the triples already treated; assign the constant function one to its additional roles. This proves Lemma 2.1 in its full stated scope.