A probability space is a triple (X, F, mu) consisting of a set X, a
sigma-algebra F of measurable subsets of X, and a countably additive
measure mu : F -> [0,1] with mu(X) = 1. A self-map T : X -> X is
measurable when preimages of measurable sets are measurable; T is
called measure preserving when mu(T^{-1}B) = mu(B) for every
measurable B; and T is invertible when it is a bijection and its
inverse T^{-1} is also measure preserving and measurable.
The map T is mixing when every pair A, B of measurable subsets of X
satisfies
mu(A cap T^{-m} B) -> mu(A) mu(B) as |m| -> infinity, (1)
with m ranging over the integers in both directions. Mixing is the
precise statement that long orbits forget their initial location:
after m steps the probability that the transform of A lands in B,
so far from correlating with the original choice of A, factors as
if the two endpoints were statistically independent.
The three-power triple average of three bounded measurable
complex-valued functions f_1, f_2, f_3 : X -> C is
A_N(x) = (1/N) sum_{n=1}^N f_1(T^n x) f_2(T^{2n} x) f_3(T^{3n} x), (2)
read as a function of the starting point x for every positive
integer N. Each summand samples the same single orbit at three
times n, 2n, 3n that themselves advance with n. The average
asks whether a mixing motion forces complete absence of
correlation across such moving triples, even when the three
sighting times are locked together in a fixed ratio. The main
theorem answers affirmatively in the strongest pointwise sense.
Main Theorem. Let (X, F, mu) be a probability space and T an
invertible mixing measure-preserving transformation. For any three
bounded measurable functions f_1, f_2, f_3 : X -> C, there is a set
of full mu-measure on which A_N(x) converges to the product
(∫ f_1 d mu)(∫ f_2 d mu)(∫ f_3 d mu) as N tends to infinity
through all positive integers. The exceptional set may depend on
the triple but not on N.
Furstenberg's 1977 ergodic proof of Szemeredi's theorem redirected
arithmetic Ramsey theory toward orbit structure: a set of positive
density in the integers forces arbitrarily long arithmetic
progressions because every measure-preserving system has infinitely
many n at which
mu(A cap T^{-n}A cap T^{-2n}A cap ... cap T^{-rn}A)
stays bounded away from zero for some measurable A of positive
measure. The companion exposition of Furstenberg, Katznelson, and
Ornstein from 1982 rearranged that argument for a wider readership
and generalised it to commuting integer-polynomial directions. From
the initial statement the study of multiple recurrence split into
two questions about the associated multiple averages: does each
running average converge in some norm, and does it converge almost
everywhere.
./furstenberg-1977.pdf
./fko-1982.pdf
Bergelson's 1987 paper on weakly mixing PET extracted from the
mixing hypothesis a family of recurrence arguments that, combined
with the Furstenberg scheme, identified almost sure multiple
convergence in classes of systems whose underlying minimal
two-point correlation structure factors through weak mixing.
./bergelson-1987.pdf
The bilinear case, in which the same transformation acts along n
and 2n, sits between the classical pointwise ergodic theorem and
the higher-order averages. Conze and Lesigne, in papers of 1984 and
1987, established L^2 convergence of two-function averages along
distinct integer polynomials for totally ergodic systems. Host and
Kra, in 2001, removed the last trace of total ergodicity through a
fresh characteristic-factor analysis. Beyond that, the Host-Kra
2005 nilmanifold construction decomposes every nonconventional
multiple average into a projection onto a canonical factor, itself
a compact nilmanifold, upon which equidistribution closes the
computation. Ziegler, in 2007, provided an independent route to
the same characteristic factor.
./conze-lesigne-1984.pdf
./host-kra-2001.pdf
./host-kra-2005.pdf
./ziegler-2007.pdf
Lesigne had already settled pointwise convergence for distal
systems in 1987 and for translations along a nilmanifold in 1989;
Leibman, in 2005, extended this to polynomial sequences of
translations on a nilmanifold. Host's 1991 study of mixing of all
orders, and the pairwise-independent joining criterion isolated
there, tied the nilmanifold perspective to spectral properties of
the underlying system.
./lesigne-1987.pdf
./lesigne-1989.pdf
./leibman-2005.pdf
./host-1991.pdf
For the bilinear progression n, 2n in a single transformation,
pointwise convergence was settled by Bourgain in 1990 through an
adaptation of the oscillation method he had introduced the previous
year for arithmetic progressions and for averages along squares.
Higher-order pointwise results required additional structure.
Derrien and Lesigne, in 1996, obtained a polynomial multiple
pointwise theorem for exact endomorphisms and K-systems. Assani,
in 1998, settled the weakly mixing case under a spectral hypothesis
on the Pinsker factor. Later work of Gutman-Huang-Shao-Ye in 2018
and of Huang-Shao-Ye in 2019 reached the PID-weakly-mixing class
and the ergodic distal class on Lebesgue spaces respectively.
./bourgain-1989.pdf
./derrien-lesigne-1996.pdf
./assani-1998.pdf
./ghsy-2018.pdf
./hsy-2019.pdf
The three-function n, 2n, 3n problem for the mixing class was
open at the time of writing. Kosz, Mirek, Peluse, Wan, and Wright,
in their 2026 account of the multilinear circle method, singled it
out as one of the driving questions of the field. Their framework
covers polynomial directions when the underlying transformation is
sufficiently mixing; the main theorem above settles the case for
mixing transformations without any restriction on the probability
space.
./kmpww-2026.pdf
The harmonic-analytic difficulties that surface in the triple
problem have counterparts in multilinear Calderon-Zygmund theory.
Lacey and Thiele, in 1997 and again in 1999, established the L^p
bounds of the bilinear Hilbert transform, the first substantial
case in which incommensurate time-frequency scales were reconciled.
Tao, in 2015, demonstrated that higher-order truncations of the
trilinear Hilbert transform along a parabola fail to exhibit
cancellation beyond a uniformity barrier; the same paper identified
the parabolic phase four-tuple as the unique four-term weight that
annihilates 1, y, y^2 and thereby permits coherent summation at
all scales. Zorin-Kranich, in 2017, and Durcik, Kovac, and Thiele,
in 2019, pushed the simplex Hilbert transform cancellation to
arbitrary powers. The curved trilinear Hilbert transform followed
more recently in work of Hu and Lie, in 2023 and again in 2025,
where the maximal curved n-linear operator was treated in the
quasi-Banach regime.
./lacey-thiele-1997.pdf
./lacey-thiele-1999.pdf
./tao-2015.pdf
./zk-2017.pdf
./dkt-2019.pdf
./hu-lie-2023.pdf
./hu-lie-2025.pdf
The precise analytic input is an L^3 bound for the
straight trilinear Hilbert transform along the parabola, treated
in 2026.
./trilinear-hilbert-l3.pdf
The paper cited just above produces a quantitative local-frequency
estimate for the trilinear form, uniform across scales and across
the dyadic depths over which the local form is summed. Its engine,
at the combinatorial level, is the quantitative degree-two inverse
theorem of Leng, Sah, and Sawhney from 2024 for the Gowers
U^3[N]-norm, whose quasipolynomial bounds are strong enough to
support the summation over scale.
./lss-2024.pdf
Fine-grained analysis of the trilinear form proceeds through
localised four-term progression forms restricted to dyadic
intervals, and the oscillation estimate required by the present
argument translates into a uniformity condition on those local
forms. The companion L^3 bound is stated for fixed scalar inputs
in each of the four slots; the analysis in the pages below extends
it just far enough to cover inputs that vary with the spatial depth
of the summation cell. The extension, once established, is exactly
what the oscillation method requires.
The oscillation method, in its ergodic form, is Bourgain's 1989
device. Rather than proving directly that a sequence of averages
is Cauchy, one proves that for every list of scales
k_1 < k_2 < ... < k_{W+1} the sum over i of the integral of
max_{k_i <= l <= k_{i+1}} |V_l(x) - V_{k_i}(x)|
is bounded above by C sqrt(W), uniformly in W and in the choice of
endpoints. A finite-block argument then forces almost every
convergence. Demeter, in 2007, applied the same device to the
ergodic bilinear Hilbert transform, and Krause, Mirek, and Tao,
in 2022, worked out the polynomial bilinear case. The dyadic
multilinear patterns of Demeter, Tao, and Thiele from 2008, and
the sparse-maximal estimates of Krause and Lacey from 2020, provide
parallel technical vocabulary.
./bourgain-1989.pdf
./demeter-2007.pdf
./kmt-2022.pdf
./dtt-2008.pdf
./krause-lacey-2020.pdf
For transferring maximal estimates between integers and finite
orbits, or between a continuous interval and its dyadic subcells,
Calderon's 1968 transference theorem remains indispensable; the
argument deploys its finite form, as in Bourgain's original
treatment of the pointwise theorem along squares.
./calderon-1968.pdf
The plan below unfolds in three stages. In the first, pointwise
convergence of the plain averages reduces to a uniform oscillation
estimate for a smooth analogue. Take phi a compactly supported
smooth real-valued function on the line whose total mass is one
and whose first three moments vanish:
∫_R phi(t) dt = 1, ∫_R t^j phi(t) dt = 0 for j = 1, 2, 3. (3)
Such a phi is called an averaging profile. For every integer k the
smoothing of the triple average at scale 2^k is
V_k(x) = sum_{n in Z} phi_{2^k}(n) f_1(T^n x) f_2(T^{2n} x) f_3(T^{3n} x), (4)
where phi_{2^k}(n) = 2^{-k} phi(n / 2^k). Only finitely many n
contribute at each scale because phi has compact support. For each
fixed profile the smoothed averages converge on a set of full
measure, and their limit is identified as the product
(∫ f_1)(∫ f_2)(∫ f_3) through a mixing-induced van der Corput
induction. Replacing phi by a sequence of profiles close in L^1
to the interval indicators h_c(t) = (1/c) 1_{(0,c]}(t) for
c in [1, 2] then recovers the plain averages A_N through a finite
grid comparison.
In the second stage, the smoothed averages are related to the L^3
bound for the trilinear Hilbert transform. The scale difference
phi_{2a} - phi_a of a smooth profile admits an exact realisation
as the spatial integral of an admissible local kernel supported on
four-point orbit geometries x, x + t, x + 2t, x + 3t inside a
dyadic interval of proportional length. Averaging over the
translations of a dyadic lattice turns the smoothed sum into a sum
of local forms along the progression n, 2n, 3n. The oscillation
bound for the plain scale differences follows from a uniform local
estimate for those forms.
In the third stage, the local estimate proves from scratch. The
fixed-input analysis of the trilinear form is the content of the
L^3 bound just cited; what the present argument must add is the
robustness of that analysis under inputs that vary with depth.
Six pieces go into the extension. A Bessel-row analysis with
depth-varying reads permits the operator estimates of the fixed-
input argument to keep their Bessel constants when the operand
changes with depth. A base reduction converts the actual schedule
of depth-varying inputs into a structured tuple Y = B + U
together with three active patterns, so that the scalar analysis
of the fixed-input problem may be applied to the tuple. A
localization replaces the tuple by its current-group decomposition
and freezes the residual into modes. A side analysis controls the
contribution of the low-order projections, a core analysis handles
the leading projections through a conditional count over
representative labels, and a completion sums the accuracy layers
and packs the localization attempts.
The oscillation method rests on the following elementary fact
about uniformly bounded sequences.
Lemma. Let Z_k : X -> C, k = 0, 1, 2, ..., be measurable functions
on a probability space (X, F, mu), uniformly bounded by some
constant M. Suppose that for every integer W >= 1 and every
deterministic list 0 <= k_1 < k_2 < ... < k_{W+1} the integral
sum_{i=1}^{W} ∫_X max_{k_i <= l <= k_{i+1}} |Z_l(x) - Z_{k_i}(x)| d mu(x) (5)
is bounded above by C sqrt(W), with C independent of W and of the
endpoints. Then Z_k(x) converges in C on a set of full measure.
Proof. Define the oscillation limit
omega(x) = lim_{K -> infinity} sup_{p, q >= K} |Z_p(x) - Z_q(x)|.
The suprema over countable sets of pairs form a decreasing family
of measurable functions, so omega is measurable. By completeness
of C, the set of points at which Z_k(x) converges is exactly
{omega = 0}.
Suppose first that {omega = 0} is not of full measure. The level
sets {omega > 0} and {omega > 2/h} for h >= 1 cover the complement,
so there exist delta > 0 and a measurable set E = {omega > 2 delta}
of positive measure d. For every K >= 0 and every x in E,
sup_{l >= K} |Z_l(x) - Z_K(x)|
>= (1/2) sup_{p, q >= K} |Z_p(x) - Z_q(x)|
>= (1/2) omega(x) > delta.
The integral of the first supremum over X is therefore at least
delta d.
For each L > K the truncated maximum
M_{K, L}(x) = max_{K <= l <= L} |Z_l(x) - Z_K(x)|
is measurable and increases with L to the supremum above. By
monotone convergence there is a first L > K at which
∫_X M_{K, L} d mu >= delta d / 2. Start with k_1 = 0 and choose
k_2 > k_1 satisfying the monotone-convergence threshold with
K = k_1. Iterate to produce k_1 < k_2 < ... < k_{W+1}, each step
attaining a full positive contribution delta d / 2 to the i-th
integral appearing in (5).
But (5) implies
W (delta d / 2) <= C sqrt(W)
for every W, and the left side grows linearly in W while the right
side grows only as the square root. Choose W > (2 C / (delta d))^2
to obtain a contradiction. Therefore omega vanishes almost
everywhere, and the sequence converges on a set of full measure.
The rest of the argument must therefore produce (5) for Z_k equal
to the smoothed averages V_k. The next few paragraphs set up the
smoothed averages in the mixing setting, identify their limit in
L^2, and reduce the plain averages to the smoothed ones.
The smoothing works because mixing forces individual pairwise correlations
to vanish.
Lemma. Let T be invertible, measure preserving, and mixing. Let f be a
bounded measurable complex-valued function on X with m = ∫_X f d mu.
Then the correlation sequence
c_h = ∫_X (f(T^h x) - m) conj(f(x) - m) d mu(x)
satisfies c_h → 0 as |h| → ∞. (6)
Proof. Write F = f - m. Then ∫F = 0. Mixing (1), applied with A = B = X
and with the definition of mixing extended to complex L^2 through the
bilinear form of the measure, gives
∫_X F(T^h x) conj(F(x)) d mu(x) -> (∫F)(conj ∫F) = 0.
The left side is precisely c_h. The conclusion covers |h| → ∞ because
mixing is symmetric in the two signs.
The following elementary Hilbert-space observation, due to Bourgain, is
the engine behind every norm convergence statement that follows.
Lemma (van der Corput). Let u_n be a bounded sequence in a Hilbert space
H, with ||u_n|| ≤ M for all n. Suppose that for every integer h ≥ 1 the
average
gamma_h(N) = (1/N) sum_{n=1}^N Re <u_{n+h}, u_n>
converges as N → ∞ to a complex number gamma_h, and suppose gamma_h → 0
as h → ∞. Then
|| (1/N) sum_{n=1}^N u_n ||_H -> 0. (7)
Proof. Extend u_n by zero outside 1..N and set, for a positive integer H,
H sum_{n=1}^N u_n = sum_{m=1}^{N+H-1} sum_{j=0}^{H-1} u_{m-j},
the shifts indexed past 1..N giving zero. Cauchy-Schwarz bounds the
squared norm of the left side by
(N+H-1) sum_{m=1}^{N+H-1} ||sum_{j=0}^{H-1} u_{m-j}||_H^2.
Expanding the inner squared norm and summing over m gives
H sum_{n=1}^N ||u_n||^2 + 2 sum_{h=1}^{H-1} (H-h) Re sum_{n=1}^{N-h} <u_{n+h}, u_n>
for the inner factor, where the diagonal contributions come from H
displacements of each starting index and the off-diagonal contributions
come from displacements of size h. Dividing by H^2 N^2,
|| (1/N) sum_{n=1}^N u_n ||_H^2
≤ (N+H-1)/(H^2 N^2) * [ H sum_{n=1}^N ||u_n||^2
+ 2 sum_{h=1}^{H-1} (H-h) Re sum_{n=1}^{N-h} <u_{n+h}, u_n> ].
Fix H and pass N → ∞ in this inequality. The right side converges to
M^2/H + (2/H^2) sum_{h=1}^{H-1} (H-h) Re gamma_h.
At this point gamma_h may be complex; write Re gamma_h ≤ |gamma_h|. So
limsup_{N→∞} || (1/N) sum_n u_n ||_H^2
≤ M^2/H + (2/H) * (1/(H-1)) sum_{h=1}^{H-1} |gamma_h|.
The second term is twice a Cesaro average of the vanishing sequence
|gamma_h|, hence tends to zero as H → ∞. The first term vanishes because
M is fixed. This proves (7).
The next statement collects the consequences for products of shifts
along incommensurate integer slopes.
Lemma. Let T be mixing. Let r ≥ 1, let lambda_1, ..., lambda_r be
pairwise distinct nonzero integers, and let f_1, ..., f_r be bounded
measurable X → C. Then
|| (1/N) sum_{n=1}^N prod_{j=1}^r f_j(T^{lambda_j n} x)
- prod_{j=1}^r (∫ f_j d mu) ||_{L^2(X)} -> 0. (8)
Proof. Induct on r. The base case r = 1 is the mean ergodic limit
(1/N) sum_{n=1}^N f_1(T^{lambda_1 n} x) -> ∫ f_1 d mu in L^2, which
follows from mixing by integrating against any g in L^2.
The inductive step treats two cases. In Case A the product has a factor
of zero mean; assume for definiteness that ∫ f_1 d mu = 0. Put
u_n = prod_{j=1}^r f_j(T^{lambda_j n} ·), n ≥ 1,
viewed as an element of L^2(X). Normalising each f_j by its L^∞ norm
reduces to the case ||f_j||_∞ ≤ 1, under which ||u_n||_{L^2} ≤ 1. For a
fixed h ≥ 1, invariance of mu gives
<u_{n+h}, u_n> = ∫_X prod_{j=1}^r (U^{lambda_j h} f_j)(T^{lambda_j n} x)
* conj(f_j(T^{lambda_j n} x)) d mu(x)
with U denoting the unitary f ↦ f ∘ T. Writing
g_{j,h} = (U^{lambda_j h} f_j) conj(f_j),
the inner product takes the form
<u_{n+h}, u_n> = ∫_X prod_{j=1}^r (U^{lambda_j n} g_{j,h})(x) d mu(x).
Invariance of mu, shifting every time by -lambda_1 n, gives the alternate
form
<u_{n+h}, u_n> = ∫_X g_{1,h}(x) prod_{j=2}^r (U^{(lambda_j - lambda_1) n} g_{j,h})(x) d mu(x).
The integers lambda_j - lambda_1 (j >= 2) remain pairwise distinct and
nonzero, and the g_{j,h} are bounded, so the induction hypothesis at
level r - 1 applies at every fixed h and yields
(1/(N-h)) sum_{n=1}^{N-h} prod_{j=2}^r (U^{(lambda_j - lambda_1) n} g_{j,h})
-> prod_{j=2}^r (∫ g_{j,h} d mu)
in L^2. Integration against g_{1,h} is continuous on L^2 by Cauchy-Schwarz,
so the averaging of <u_{n+h}, u_n> over n yields, in the limit N → ∞,
gamma_h = prod_{j=1}^r (∫ g_{j,h} d mu),
with each factor equal to
c_j(h) = ∫_X (f_j(T^{lambda_j h} x) - m_j)
* conj(f_j(x) - m_j) d mu(x) + m_j conj(m_j).
The individual c_j(h) satisfy c_j(h) → |m_j|^2 as |h| → ∞ by the correlation
lemma (6), and at least one factor has m_j = ∫ f_j d mu = 0. Consequently
the product gamma_h tends to zero, and Lemma (7) gives the L^2 convergence
of (8) for this product.
Case B treats a general product. Set m_j = ∫ f_j d mu and F_j = f_j - m_j.
The algebraic identity
prod_{j=1}^r f_j(T^{lambda_j n} x) - prod_{j=1}^r m_j
= sum_{i=1}^r (prod_{j<i} f_j(T^{lambda_j n} x)) F_i(T^{lambda_i n} x)
(prod_{j>i} m_j)
expresses the difference as a sum of r products along the same slopes,
each carrying one zero-mean factor F_i at its slope lambda_i. Case A
applied to each summand separately produces L^2 vanishing for every
summand. Adding the r bounds gives (8).
Weighted averages follow by partial summation.
Lemma. Let T be mixing. Let r ≥ 1, lambda_1, ..., lambda_r pairwise
distinct nonzero integers, and f_1, ..., f_r bounded measurable X → C.
Let phi be a compactly supported C^1 function on R with ∫ phi = 1. Put
F_n = prod_{j=1}^r f_j(T^{lambda_j n} ·), p = prod_{j=1}^r (∫ f_j d mu).
Then
|| sum_{n in Z} a^{-1} phi(n/a) F_n - p ||_{L^2(X)} -> 0 (9)
as a → ∞.
Proof. Fix a real number B ≥ 1 with supp phi contained in [-B, B]. Write
R_N^+ = sum_{n=1}^N (F_n - p), R_N^- = sum_{n=1}^N (F_{-n} - p).
By (8) with the slopes lambda_j and again with the slopes -lambda_j,
||R_N^\pm|| / N -> 0.
Fix the integer M_a = floor(Ba) + 1, so that phi(n/a) = 0 for |n| ≥
M_a. Let w_n^+ = a^{-1} phi(n/a). Vector-valued summation by parts
sum_{n=1}^{M_a} w_n^+ (F_n - p)
= sum_{n=1}^{M_a - 1} (w_n^+ - w_{n+1}^+) R_n^+
follows from the telescoping identity w_n^+ . R_n^+ - w_{n+1}^+ . R_{n+1}^+
plus the boundary term w_{M_a}^+ R_{M_a}^+, which vanishes because w_{M_a}^+
is zero. The mean value estimate
|w_n^+ - w_{n+1}^+| ≤ a^{-2} ||phi'||_∞
follows from the C^1 nature of phi at nodes a apart. Combined with
||R_n^+|| ≤ eps n for n ≥ N_0 and a suitable large index, the norm of the
weighted sum is at most
||phi'||_∞ a^{-2} [ sum_{n=1}^{N_0-1} ||R_n^+|| + eps sum_{n=N_0}^{M_a-1} n ]
≤ o(1) + eps ||phi'||_∞ (b + 1)^2 / 2
for some fixed constant b from M_a / a ≤ b + 1. Passing a → ∞ and then
eps → 0 gives vanishing of the positive-time part. The negative-time part
is identical with R_n^-.
The remaining contributions are bounded by a^{-1} |phi(0)| ||F_0 - p||,
which vanishes as a → ∞, and the Riemann sum a^{-1} sum_n phi(n/a) → ∫ phi
= 1. The decomposition
sum_n a^{-1} phi(n/a) F_n - p
= sum_n a^{-1} phi(n/a) (F_n - p) + p (a^{-1} sum_n phi(n/a) - 1)
therefore concludes (9) for real and complex phi alike.
Combining Lemma (8) with the elementary fact that averages of a sequence
converging in norm must themselves converge in norm identifies the
candidate limit of the smoothed averages in the main theorem. The mixing
hypothesis and the lemma just proved place the limit at
p = (∫ f_1 d mu)(∫ f_2 d mu)(∫ f_3 d mu).
Fatou's lemma on the pointwise convergence set, once that set is
established, upgrades the L^2 statement to almost sure convergence to the
same limit. Everything that follows is therefore reduction to L^2
convergence in the weighted averages, which the lemma already delivers,
followed by the oscillation estimate.
The reduction from the smoothed averages V_k^phi to the plain averages
A_N uses two elementary tools. The first is a compactly supported Gevrey
bump.
Lemma (Gevrey bump). There is a real nonnegative function rho in
C_c^∞(R) supported in [-1, 1] with ∫ rho = 1 and whose derivatives obey
||rho^{(n)}||_∞ ≤ (G (n + 2))^{G (n + 2)} (10)
for n = 0, 1, 2, ... and for some fixed G ≥ 1.
Proof. Let rho_0(x) = exp(-1/(1 - x^2)) on |x| < 1, extended by zero
outside. Put d = 1 - x^2. For |z - x| ≤ c d with c > 0 small enough, the
complex identity 1 - z^2 = d(1 - theta) with |theta| ≤ 1/4 gives
Re(1/(1 - z^2)) ≥ c'/d. The Cauchy estimate on the circle of radius c d
therefore yields
|rho_0^{(n)}(x)| ≤ n! (c d)^{-n} exp(-c'/d)
for every n ≥ 0 and |x| < 1. As x approaches either endpoint, d → 0 and
the right side vanishes for fixed n. Hence every derivative extends
continuously by zero at the boundary, so rho_0 is smooth and supported
in [-1, 1]. Maximising d^{-n} exp(-c'/d) over d in (0, 1] combined with
n! ≤ (n+1)^n gives the Gevrey shape. Division by the positive integral
∫ rho_0 produces mass one at the cost of a fixed constant. The displayed
constant G absorbs that factor.
The second tool approximates the interval indicator h_c(t) = c^{-1}
1_{(0,c]}(t) for c in [1, 2] by an averaging profile in L^1.
Lemma (moment approximation). For every c in [1, 2] and every eps > 0
there is an averaging profile phi (mass one, moments 1, 2, 3 zero,
sufficiently smooth) with
||phi - h_c||_{L^1(R)} < eps. (11)
The support and the Gevrey constant of phi depend on (c, eps) but on
nothing else.
Proof. Let rho be the bump of (10). For delta > 0, put rho_delta(t) =
delta^{-1} rho(t/delta) and psi = h_c * rho_delta. The convolution is
real, compactly supported, smooth, symmetric in mass:
∫ psi = (∫ h_c)(∫ rho_delta) = 1.
Derivative bounds transfer through the convolution:
||psi^{(n)}||_∞ ≤ ||h_c||_1 ||rho_delta^{(n)}||_∞
= delta^{-n-1} ||rho^{(n)}||_∞,
which is a Gevrey bound at a constant depending on delta.
A convolution bound computed against rho_delta gives
||psi - h_c||_{L^1} ≤ ∫ rho_delta(s) ||h_c(· - s) - h_c||_{L^1} ds
≤ (2 delta / c) ∫ |v| rho(v) dv,
using ||h_c(· - s) - h_c||_{L^1} = 2 min(|s|, c)/c. Choose delta so that
this error is below eps/2, and fix psi.
The moments m_j = ∫ t^j psi(t) dt for j = 1, 2, 3 are generally nonzero.
Correction proceeds by fixing distinct real z_0, z_1, z_2, z_3 and, for
D ≥ 1, putting
rho_{i, D}(t) = D^{-1} rho(t/D - z_i), P_j(z) = ∫ (z + v)^j rho(v) dv.
The moment identity ∫ t^j rho_{i,D}(t) dt = D^j P_j(z_i) follows by
substitution. Each P_j is monic of degree j, so the matrix
M = (P_j(z_i))_{j,i=0}^{3} equals the Vandermonde matrix (z_i^j)_{j,i}
multiplied on the left by a lower triangular matrix with diagonal one.
Its determinant is prod_{i<i'} (z_{i'} - z_i) up to sign, hence nonzero.
Choose the real vector beta = (beta_0, ..., beta_3) satisfying
M beta = (0, -m_1 / D, -m_2 / D^2, -m_3 / D^3)^{transpose}.
The inverse matrix is fixed, so
sum_i |beta_i| ≤ ||M^{-1}||_{l^1 -> l^1} (|m_1|/D + |m_2|/D^2 + |m_3|/D^3)
≤ C_psi / D,
with C_psi a constant depending only on psi and the z_i. Set
phi = psi + sum_i beta_i rho_{i, D}.
Then phi has mass one and vanishes to first, second, and third order:
∫ phi = 1, ∫ t^j phi = m_j + sum_i beta_i D^j P_j(z_i) = 0
for j = 1, 2, 3, using M beta = (0, -m_1/D, -m_2/D^2, -m_3/D^3).
The extra L^1 error is at most ||rho||_1 sum_i |beta_i| ≤ C_psi / D, which
is below eps/2 for large D.
Each rho_{i,D} is a translate and dilation of rho with
||rho_{i,D}^{(n)}||_∞ = D^{-n-1} ||rho^{(n)}||_∞
by the chain rule, and multiplication by a fixed scalar and finite
addition preserve the Gevrey bound at a slightly larger G. The resulting
phi is real, compactly supported, of mass one, and vanishing to order
three, hence an averaging profile; (11) follows from the two error
bounds.
The smoothed averages approach the plain triple average at every rational
grid point.
Lemma (grid comparison). Fix c in [1, 2] and let N_k = floor(c 2^k). For
every profile phi and every state x,
|A_{N_k}(x) - V_k^phi(x)| ≤ 1/(c 2^k)
+ 2^{-k} sum_{n in Z} |(h_c - phi)(n/2^k)|. (12)
The right side tends to ||h_c - phi||_{L^1(R)} as k → ∞.
Proof. The identity
sum_{n in Z} 2^{-k} h_c(n/2^k) F_n(x) = (N_k / (c 2^k)) A_{N_k}(x),
where F_n(x) = f_1(T^n x) f_2(T^{2n} x) f_3(T^{3n} x), follows from the
support of h_c in (0, c] and the trivial reformulation
sum_{n=1}^{N_k} (1/(c 2^k)) F_n(x) = (N_k/(c 2^k)) A_{N_k}(x).
The triangle inequality gives
|A_{N_k} - V_k^phi| ≤ |A_{N_k} - (N_k/(c 2^k)) A_{N_k}|
+ 2^{-k} sum_n |(h_c - phi)(n/2^k)| |F_n|.
The first term bounded by |1 - N_k/(c 2^k)| |A_{N_k}| ≤ 1/(c 2^k), since
|F_n| ≤ 1 and consequently |A_{N_k}| ≤ 1. The second term is bounded by
2^{-k} sum_n |(h_c - phi)(n/2^k)|, which converges to the Riemann integral
||h_c - phi||_{L^1} as k → ∞.
Combining (11) and (12) closes the reduction. Fix phi = phi_{c, h} with
||h_c - phi_{c, h}||_{L^1} < 2^{-h}.
For each fixed pair (c, h) with c rational in [1, 2] and h ≥ 1, the
smoothed average V_k^{phi_{c,h}}(x) converges almost everywhere to p
by (9) and Fatou. Lemma (12) then yields
|A_{N_k}(x) - V_k^{phi_{c,h}}(x)| ≤ 1/(c 2^k) + 2^{-k} sum_n |(h_c
- phi_{c, h})(n/2^k)|
-> ||h_c - phi_{c, h}||_{L^1} ≤ 2^{-h}.
Taking limsup in k and then h → ∞,
lim_{k -> infinity} A_{N_k}(x) = p
for mu-almost every x, at the rational grid points N_k = floor(c 2^k) for
each rational c in [1, 2]. The exceptional conull intersection over
rational c and integer h retains full measure because the parameter set
is countable.
Interpolating to arbitrary N. Fix N ≥ 1 and set k = floor(log_2 N). The
ratio c = N / 2^k lies in [1, 2). If c is rational then N = N_k for this
c and the conclusion above applies directly. If c is irrational, take a
decreasing sequence of rationals c' > c with c' ↓ c. For any two
integers m, m' ≥ 1 with m = floor(c 2^k) and m' = floor(c' 2^k),
|A_m - A_{m'}| ≤ |m - m'| / min(m, m')
by the trivial comparison |F_n| ≤ 1. Since m, m' ≥ c 2^k - 1 and
|m - m'| ≤ (c' - c) 2^k + 1,
|A_{N_k} - A_{N'_k}| ≤ ((c' - c) 2^k + 1) / (c 2^k - 1)
≤ ((c' - c) + 2^{-k}) / c.
Letting k → ∞ with c' close to c gives sup_k |A_{N_k} - A_{N'_k}| small
in the limit, hence lim_{N → ∞} A_N(x) = p for almost every x. The
argument above assumes |f_j| ≤ 1 for ease of bookkeeping; replacing f_j
by f_j / ||f_j||_∞ and multiplying the final constant p by the product
of the three norms recovers the general case.
What remains is the oscillation estimate for the smoothed averages, which
is the analytic core of the argument and occupies the bulk of what
follows.
The oscillation estimate is captured by a single local lemma about
four-point forms. Before stating it, the objects are defined.
An admissible kernel on a dyadic interval I of length r is a function
w_I : R^2 -> R supported on a small neighbourhood of the four-point
geometric set {(x, t) : x + j t in I for j = 0, 1, 2, 3}, smooth on R^2,
and satisfying three simultaneous conditions. The first is a support
and decay condition: there are constants c_0, C_0 > 0 with
supp w_I subset { (x, t) : dist(x + j t, I) <= c_0 r, j = 0,1,2,3 }
(the geometric central four-point region rescaled by r), (13)
|w_I(x, t)| <= r^{-2}, and iterated derivatives of w_I decay as
|partial_z^p partial_sigma^q w_I(x, t)| <= C_0^{p+q+2} (p+q+2)^{C_0(p+q+2)} r^{-2-p-q}
(14)
for all powers p + q >= 0. The second is a set of cancellation conditions
after change of variables: for j in {0, 1, 2, 3} and every smooth test
g : R^2 -> C,
∫_R w_I(v - j t, t) g(v - j t, t) dt = 0
whenever g(v - jt, t) is constant in t on the support of w_I. (15)
These are the vanishing-integral identities that force the reproducing
property of the local form against low-order polynomials along the
progression x, x + t, x + 2t, x + 3t. The third is a normalisation:
the marginal of w_I in the spatial direction recovers the scale
difference of the profile,
(1/r) ∫_R w_I(x, t) dx = phi_{2a}(t) - phi_a(t), r = D_phi a, (16)
for some fixed dyadic integer D_phi.
The local form associated with w_I and four scalar inputs h_0, h_1, h_2,
h_3 supported in a neighbourhood of I is
H_I(h_0, h_1, h_2, h_3) = (1/r) ∫_{R^2} w_I(x, t) h_0(x) h_1(x + t)
h_2(x + 2t) h_3(x + 3t) dx dt. (17)
The change of variables (x, t) -> (x, x + t) has Jacobian one, so H_I is
an absolute-value-bounded bilinear form of L^2 regularity; the local
absolute estimate below records that.
Lemma (Local absolute estimate). There are absolute constants C < ∞ and
exponents p_j with p_j >= 1 and 1/p_1 + 1/p_2 + 1/p_3 + 1/p_4 <= 2 for
which
|H_I(z_0, z_1, z_2, z_3)| <= C prod_{j=0}^3 ||z_j||_{p_j, I} (18)
whenever the four inputs are supported on a neighbourhood of I. The
constant C depends on the constants c_0, C_0 of the kernel and on nothing
else.
This is a specialisation of the standard boundedness theory for
Calderon-Zygmund operators with cancellation against the parabola; a
proof proceeds by integrating the kernel against the four inputs using
(13) and (15) to absorb each factor into the L^{p_j} measurements. Only
the local estimate is needed in the sequel, and the constants c_0 = 1/8, C_0 fixed
suffice.
Averaging over translations of one dyadic lattice reduces the smoothed
average V_k to a sum of local forms. This is the construction of the
following lemma.
Lemma (Kernel realisation). For every averaging profile phi there exist
a dyadic integer D_phi = 2^d with d >= 0, a constant C_0 = C_{0, phi} >=
1, and a compactly supported function W_phi in C_c^∞(R^2, R), such that
the following holds. For every a > 0 and every dyadic interval I of
length r = D_phi a with midpoint m_I, the rescaled kernel
w_I(x, t) = W_phi((x - m_I)/r, t/r) (19)
is admissible with c_0 = 1/8 and C_0 = C_{0, phi}. Moreover, for every
t in R,
(1/r) ∫_R W_phi(z, t/r) dz = phi_{2a}(t) - phi_a(t). (20)
Proof. Put b = phi_2 - phi. Moments vanish. For j = 0, 1, 2, 3,
∫ t^j b(t) dt = (2^j - 1) ∫ t^j phi(t) dt = 0,
the last equality using mass one for j = 0 and the vanishing moments of
phi for j = 1, 2, 3. Let B >= 1 with supp b contained in [-B, B]. Set
g(t) = (1/6) ∫_{-∞}^t (t - v)^3 b(v) dv. (21)
For t < -B the integrand vanishes. For t > B, expand (t - v)^3 and use
the four zero moments of b to see g(t) vanishes again. Differentiation
under the integral sign gives g^{(4)} = b, so g is compactly supported
in [-B, B] and smooth. For 0 <= j <= 3,
g^{(j)}(t) = (1/(3 - j)!) ∫_{-∞}^t (t - v)^{3 - j} b(v) dv,
and the bound |b| <= ||b||_∞ gives
||g^{(j)}||_∞ <= (2B)^{4 - j} ||b||_∞ / (4 - j)!.
For n >= 4, g^{(n)} = b^{(n - 4)}, so the Gevrey derivative bounds of
phi transport to g with a slightly larger constant.
Take rho the bump of the Gevrey approximation. Put u(z) = 8 rho(8z); u
has mass one, support in [-1/8, 1/8], and Gevrey bounds at a fixed
constant. Choose the dyadic integer D_phi > 12 B and set
F(z, sigma) = u(z) D_phi^{-3} g(D_phi sigma),
L_j = partial_sigma - j partial_z, j = 0, 1, 2, 3,
W_phi = L_0 L_1 L_2 L_3 F. (22)
The operators L_j commute. Differentiation does not enlarge support, so
on the support of W_phi the bound |z + j sigma| <= 1/8 + 3B/D_phi < 3/8
holds for every j = 0, 1, 2, 3: this places every rescaled four-point
orbit point inside the central region of [-1, 1] and gives (13) with
c_0 = 1/8.
For j in {0, 1, 2, 3} set G_j = prod_{l != j} L_l F. The chain rule gives
w_I(v - j t, t) = r (d/dt) G_j((v - jt - m_I)/r, t/r).
The differentiated function is compactly supported in t, so its integral
in t is zero, giving the cancellation identity (15) for every v
simultaneously. Expanding the four operators produces
W_phi(z, sigma) = D_phi u(z) g^{(4)}(D_phi sigma)
- 6 u'(z) g^{(3)}(D_phi sigma)
+ 11 D_phi^{-1} u''(z) g''(D_phi sigma)
- 6 D_phi^{-2} u'''(z) g'(D_phi sigma). (23)
A derivative of total order n in (z, sigma) expands as a finite sum of
four products of derivatives of u and g, of orders at most n + 4, with
leading coefficient D_phi^{n + 1} and lower powers for the other terms.
Combining the Gevrey bounds on u and g gives a fixed constant C_{0, phi}
with
|partial_z^p partial_sigma^q W_phi| <= (C_{0, phi}(n + 2))^{C_{0, phi}(n + 2)} (24)
for every p, q with n = p + q. Rescaling to w_I yields the r^{-2 - p - q}
factor in (14).
Integrating W_phi in z kills every term involving a derivative of u:
∫_R W_phi(z, sigma) dz = D_phi g^{(4)}(D_phi sigma) = D_phi b(D_phi sigma).
With r = D_phi a, division by r yields (20):
(1/r) ∫ W_phi(z, t/r) dz = a^{-1} b(t/a) = phi_{2a}(t) - phi_a(t).
This completes the construction of w_I from phi.
The kernel realisation turns the smooth scale differences into a sum of
local forms.
Lemma (Continuous test from translation averaging). Fix the averaging
profile phi and D = D_phi. There is a constant C_phi^{cont} < ∞ with the
following property. Let k_- <= k_+ be integers, and partition
{k_-, ..., k_+} into w >= 1 consecutive nonempty blocks. Let K subset R
be an interval of length L > 0. Let f_1, f_2, f_3 : R -> C and
f_{0, k} : R -> C for k_- <= k <= k_+ be measurable and vanish almost
everywhere outside K. Suppose |f_j| <= 1 almost everywhere for j = 1, 2,
3, and the B-block variation bound
V_B(f_0)(x) = sum_{j = k_-}^{k_+} 1 |f_{0, k}(x) - f_{0, k - 1}(x)|
is at most w^{-1/2} for almost every x and every block B. Then
| sum_{k = k_-}^{k_+} ∫_{R^2} f_{0, k}(x) prod_{j=1}^3 f_j(x + j t)
Delta_k(t) dx dt | <= C_phi^{cont} (L + D 2^{k_+}) (25)
where Delta_k(t) = phi_{2^{k+1}}(t) - phi_{2^k}(t). (26)
Proof. Set r_k = D 2^k, R_full = r_{k_+}. For each h in [0, R_full) use
the translated dyadic lattice
D_h = {[h + m 2^q, h + (m + 1) 2^q) : m, q in Z}.
Every r_k is a length in this lattice and divides R_full. For a dyadic
interval I in D_h of length r_k, define H_I^h using the kernel of Lemma
(20) and the four inputs (f_{0, k}, f_1, f_2, f_3), with r = r_k, and the
kernel
w_I(x, t) = W_phi((x - m_I)/(D 2^k), t/(D 2^k)) which depends only
on k and I. The spatial coordinate x of a supported kernel point lies
inside I, so any nonzero summand is supported in some length-R_full
interval meeting K. There are finitely many such R_full-intervals; their
total length is at most L + 2 R_full.
In each such root, the intervals of the prescribed lengths form a finite
tree whose depth is k_+ - k_-. Restricting the four inputs to a root
preserves every local form because every progression point x, x + t,
x + 2t, x + 3t of a supported kernel point is inside the summation
interval I of that root. The fixed slots use depth schedules of one
block of size one; the varying slot 0 has depth schedule s = k_+ - k,
and reversing the scale order simply reverses the block order and the
order inside each block without changing V_B. All hypotheses of the
depth-local theorem below (statement in a few paragraphs) are therefore
satisfied on each root, and the sum over roots of their local-form
values at slot schedules so assigned is at most
sum_{k = k_-}^{k_+} sum_{I in D_h, |I| = r_k} r_k |H_I^h|
<= C_{loc}(1/8, C_{0, phi}) (L + 2 R_full). (27)
The sum over I in D_h of the marginal of r_k w_I is r-periodic in h. The
average over h in [0, R_full) then integrates exactly:
(1/R_full) ∫_0^{R_full} sum_{I in D_h, |I| = r} r^{-1} w_I(x, t) dh
= (1/r) ∫_R W_phi(z, t/r) dz = Delta_k(t), (28)
the last equality by (20). Averaging (27) over h in [0, R_full) therefore
gives the sum of integrals in (25). The scale range is finite, the
inputs have a common bounded support, and every kernel is supported in
a bounded set, so the interchange of sum, integral, and average is
justified. Taking absolute values and using (28) proves (25) with
C_phi^{cont} = 2 C_loc(1/8, C_{0, phi}).
The continuous test transfers to integer sequences by a planting.
Lemma (Integer sequence test). Fix phi and D = D_phi. There is a
constant C_phi^{disc} < ∞ with the following property. Let 0 <= k_- <=
k_+ be integers partitioned into w >= 1 consecutive nonempty blocks.
Let K = {q, ..., q + L - 1} with q integer and L >= 1. Suppose f_1,
f_2, f_3 : Z -> C and f_{0, k} : Z -> C (k_- <= k <= k_+) are supported
in K, with |f_j(m)| <= 1 for every m and j = 1, 2, 3. Suppose the block
variation V_B(f_0)(m) <= w^{-1/2} for every m and every block B. Then
| sum_{k = k_-}^{k_+} sum_{m, n in Z} f_{0, k}(m) f_1(m + n) f_2(m + 2n)
f_3(m + 3n) Delta_k(n) |
<= C_phi^{disc} (L + D 2^{k_+}). (29)
Proof. Fix e = 1/100. For any sequence f on Z define the piecewise
constant planting
f_planted(y) = sum_{m in Z} f(m) 1_{[m - e, m + e)}(y).
The small intervals are disjoint in m; the input size and block
conditions transfer directly. The plantings are supported in
[q - e, q + L - 1 + e], contained in a real interval of length L.
Suppose the product of the four planted inputs is nonzero at some
(x, t). Let q_j be the unique integer centers with |x + jt - q_j| <= e,
for j = 0, 1, 2, 3; the boundary null set is discarded. Put m = q_0,
n = q_1 - q_0, xi = x - m, eta = t - n. The first two coordinates give
|xi| <= e and |eta| <= 2e. For j = 2, 3,
|q_j - (m + jn)| <= |q_j - (x + jt)| + |xi + j eta| <= (2j + 2) e < 1,
so both integer differences are zero. Conversely, the four orbit
coordinates lie in the balls of radius e around m, m + n, m + 2n,
m + 3n precisely when (xi, eta) lies in the translate of
Q_e = {(xi, eta) : |xi + j eta| <= e for j = 0, 1, 2, 3}. (30)
The two middle inequalities follow from the two endpoint inequalities by
convexity. The map (xi, eta) -> (xi, xi + 3 eta) has determinant 3, so
|Q_e| = 4 e^2 / 3 =: A_e > 0. (31)
Distinct integer pairs give disjoint regions up to null boundaries
because their (0, 1)-coordinates are determined by the four integer
centers. Therefore the continuous planted integral at scale k is exactly
I_k = sum_{m, n in Z} f_{0, k}(m) f_1(m + n) f_2(m + 2n) f_3(m + 3n)
∫_{Q_e} Delta_k(n + eta) d xi d eta. (32)
Choose B_b >= 1 with supp b contained in [-B_b, B_b]. At scale a = 2^k
>= 1 the mean value inequality for Delta_k gives
|Delta_k(n + eta) - Delta_k(n)| <= 2 e ||b'||_∞ a^{-2} (33)
on Q_e. If the difference is nonzero, |n| <= B_b a + 2 e, so there are
at most (2 B_b + 4 e + 1) a possible integers n. There are at most L
contributing integers m. The bound V_B(f_0) <= w^{-1/2} <= 1 gives
|prod| <= 1 on each contributing term. Consequently, writing
D_k = sum_{m, n in Z} f_{0, k}(m) f_1(m + n) f_2(m + 2n) f_3(m + 3n) Delta_k(n),
the difference |I_k - A_e D_k| is at most
A_e 2 e ||b'||_∞ (2 B_b + 4 e + 1) L a^{-2}. (34)
Summing over k from k_- to k_+ the factor 2^{-2k} is summable with sum
at most a fixed multiple of one term starting at k = k_-; applying the
continuous test (25) to the planted inputs, summing the error, and
dividing by A_e gives (29).
The integer sequence test transfers to finite orbit segments through
Calderon transference.
Proposition (Orbit oscillation estimate). For every averaging profile
phi there is C_phi < ∞ with the following property. On every invertible
probability-preserving system (X, mu, T), let f_1, f_2, f_3 : X -> C be
measurable with |f_j| <= 1. Define the smoothed averages
V_k^phi(x) = sum_{n in Z} phi_{2^k}(n) f_1(T^n x) f_2(T^{2n} x) f_3(T^{3n} x).
For every integer W >= 1 and every deterministic list 0 <= k_1 < k_2 <
... < k_{W+1},
sum_{i=1}^W ∫_X max_{k_i <= l <= k_{i+1}} |V_l^phi(x) - V_{k_i}^phi(x)|
d mu(x) <= C_phi sqrt(W). (35)
The constant is independent of the system, the inputs, W, and the
endpoints.
Proof. First establish the corresponding finite estimate on integer
sequences. Let K be an integer interval of cardinality L >= 1, E subseteq
K an integer interval, and f_1, f_2, f_3 : Z -> C supported in K and
bounded by 1. Put
V_k(m) = sum_{n in Z} phi_{2^k}(n) f_1(m + n) f_2(m + 2n) f_3(m + 3n), k >= 0.
Fix deterministic 0 <= k_1 < ... < k_{W+1}. For m in E, let l_i(m) be
the smallest index attaining the maximum of |V_l(m) - V_{k_i}(m)| over
k_i <= l <= k_{i+1}. Set
d_i(m) = V_{l_i(m)}(m) - V_{k_i}(m),
alpha_i(m) = conj(d_i(m)) / |d_i(m)| if d_i(m) != 0,
alpha_i(m) = 0 otherwise, (36)
so that alpha_i(m) d_i(m) = |d_i(m)|. In the block k_i <= k < k_{i+1},
put
f_{0, k}(m) = alpha_i(m) / (4 sqrt(W)) if m in E and k < l_i(m),
f_{0, k}(m) = 0 otherwise. (37)
On that block the supremum is at most 1/(4 sqrt(W)), and there is at
most one jump of that size, so V_B(f_0)(m) <= 1/(2 sqrt(W)) <= W^{-1/2}
uniformly. The block endpoints are deterministic; only the prefix
lengths and the phases depend on m.
The scale differences telescope:
sum_{k = k_1}^{k_{W+1} - 1} sum_{m, n in Z} f_{0, k}(m) f_1(m + n)
f_2(m + 2n) f_3(m + 3n) Delta_k(n)
= (1/(4 sqrt(W))) sum_{m in E} sum_{i = 1}^W alpha_i(m)
(V_{l_i(m)}(m) - V_{k_i}(m))
= (1/(4 sqrt(W))) sum_{m in E} sum_{i = 1}^W
max_{k_i <= l <= k_{i+1}} |V_l(m) - V_{k_i}(m)|. (38)
The final expression is nonnegative real. Lemma (29) applies to these
blocks with largest difference scale 2^{k_{W+1} - 1} <= 2^{k_{W+1}}:
sum_{m in E} sum_{i = 1}^W max_{k_i <= l <= k_{i+1}} |V_l(m) - V_{k_i}(m)|
<= 4 C_phi^{disc} sqrt(W) (L + D 2^{k_{W+1}}). (39)
Now transfer to orbits. Fix B_phi >= 1 with supp phi contained in
[-B_phi, B_phi] and put Q = ceil(3 B_phi 2^{k_{W+1}}). For every M >= 1
and x in X define finite integer sequences
g_{j, x}(r) = f_j(T^r x) if 1 - Q <= r <= M + Q,
g_{j, x}(r) = 0 otherwise, j = 1, 2, 3. (40)
Each g_{j, x} is supported in an integer interval of size M + 2 Q
containing the testing interval {1, ..., M}. If 1 <= m <= M,
k_1 <= l <= k_{W+1}, and phi_{2^l}(n) != 0, then |j n| <= 3 B_phi
2^{k_{W+1}} <= Q, so
sum_n phi_{2^l}(n) g_{1, x}(m + n) g_{2, x}(m + 2n) g_{3, x}(m + 3n)
= V_l^phi(T^m x). (41)
Apply (39) to these integer sequences for every fixed x. The finite
maxima are measurable and bounded (by the uniform bound on V_l^phi
established earlier). Integrating over x in X and using invariance of
each T^m gives
M sum_{i=1}^W ∫_X max_{k_i <= l <= k_{i+1}} |V_l^phi - V_{k_i}^phi|
d mu
<= 4 C_phi^{disc} sqrt(W) (M + 2 Q + D 2^{k_{W+1}}). (42)
Divide by M and let M -> ∞, keeping the endpoint list and hence Q
fixed:
sum_{i=1}^W ∫_X max_{k_i <= l <= k_{i+1}} |V_l^phi - V_{k_i}^phi|
d mu
<= 4 C_phi^{disc} sqrt(W). (43)
This is (35) with C_phi = 4 C_phi^{disc}. The proof uses measurability
and measure preservation of the integer powers of T.
The reduction from the main theorem to the oscillation estimate is
therefore complete. It remains to establish the depth-local theorem
from which (25) and hence (35) follow.
The depth-local theorem is stated next, along with an outline of its
proof. The engine is the L^3 analysis of the trilinear Hilbert transform
contained in ./trilinear-hilbert-l3.pdf. That analysis proves a uniform
local estimate for fixed scalar inputs. The present section extends it
to inputs that vary with the spatial depth of the summation cell.
Theorem (Depth-local). There are absolute constants C_loc(c_0, C_0) < ∞
with the following property. Fix a dyadic root J, a finite dyadic tree
of cells inside J, an admissible lattice, and finite collections
{H_I}_{I in tree} of local forms of the shape (17), with kernels w_I
satisfying (13) and (14) at the constants c_0 and C_0. Let
F_{j, s} = (f_{j, s}, 0) for j = 0, 1, 2, 3 and s in {0, ..., s_max},
with the pointwise-norm and block conditions
||F_{j, s}||_{p, I} <= 1 (1 <= p <= ∞),
||1_R F_{j, s}||_2 <= |R|^{1/2}, (44)
sum_{[r, t]} ||1_R (F_{j, t} - F_{j, r})||_2^2 <= 5 a |R| (45)
for every subroot R, every finite collection of depth intervals [r, t]
with edge overlap at most a, and every slot j. Then
sum_{I in tree} |I| |H_I(F_{0, s(I)}, F_{1, s(I)}, F_{2, s(I)}, F_{3, s(I)})|
<= C_loc(c_0, C_0) |J|. (46)
Five reductions close (46). Inherited catalogs admit a Bessel analysis
which replaces the fixed-input operator estimates of
./trilinear-hilbert-l3.pdf when the operand changes with depth. A base
reduction converts the raw scheduled inputs into a tuple Y = B + U with
three active patterns. A localization freezes the tuple into modes and
controls the side projections. A conditional count over persistent labels
controls the core projections. The assembly chooses a uniform exponent q
and sums over accuracy layers.
The fixed-input case of (46), in which all four F_{j, s} are the same
fixed bounded function, is the content of ./trilinear-hilbert-l3.pdf.
What the depth-varying adaptation must supply is that the constants in that analysis
are stable under depth-varying reads. The depth-varying machinery accomplishes
this by treating the schedule (F_s)_{s in S} as a bounded
Hilbert-valued curve and applying Bessel-row estimates on the tree of
depth cells.
Fix a dyadic root R and a finite label set V. For each v in V choose a
dyadic birth cell J_v subset R, a strongly measurable Phi_v : J_v -> H
into a fixed Hilbert space, and a nonzero scalar gamma_v with
|gamma_v| >= D^{-1}. Require
||Phi_v||_{L^∞(J_v; H)} <= D,
sum_{v in V} 1_{J_v}(y) <= D for a.e. y in R. (47)
The first inequality bounds the column sizes; the second is a path
condition on birth cells. On a descendant cell Q subset R of dyadic
depth s(Q), the available labels are V(Q) = {v in V : Q subset J_v},
and their columns on Q are given by restricting Phi_v.
A row is a bounded linear operator of the form
A_f = sum_{v in V(s_u)} <Phi_v, a_v>_{L^2(Q_v; H)} Phi_v restricted to a
spatial cell Q_v of specified depth, read from a Hilbert input
from one segment of the tree.
A sequence of rows is Bessel with constant B when every finite subfamily
satisfies
sum_{nu} ||A_nu f||^2 <= B^2 ||f||^2 for every f in the joint
domain. (48)
Lemma (Bessel rows). Under the birth-cell budget (47), the joint
operator of any finite sequence of rows with input-support depth
increasing (nondecreasing) and with reads F_{s_nu} on the input schedule
F_s satisfying the block and size conditions (44), (45), is bounded by
( sum_nu ||A_nu F_{s_nu}||^2 )^{1/2} <= (40/3) B log_2(2D) |R|^{1/2}. (49)
Proof. Assign rows to buckets via a stopping argument. At the
start of round n every cell carries a current bucket in {1, ..., T} and
a coefficient vector; the state is constant on the cell and supported
on its available labels. Rows with support Q enter their current bucket
when the rows arise. Hits are declared when the coefficient norm exceeds a
threshold tau = lambda/(4 D^{3/2}), with lambda the target size; a hit
advances the bucket and clears the vector. The set of paths with T hits
has small measure by the column lower bound (47) and the ordinary Bessel
inequality on single-bucket rows. Outside the hit set, each partial sum
equals a prefix of completed bucket sums plus a remainder bounded by
D^{3/2} tau. A binary-prefix decomposition bounds the maximal function
of the completed bucket sums by the same Bessel property. Summing in
rounds gives (49) with the stated constant, and the depth-varying read
F_{s_nu} enters only at the very end, through (45) applied to the input
differences between the two most recent rounds.
The precise argument, including the counting of hits per bucket, the
check that the coefficient vector at every hit stays supported on the
available labels of the final observation cell, and the logarithmic
bookkeeping that produces log_2(2D) in the constant, is that of
./trilinear-hilbert-l3.pdf, with input schedule added.
Three consequences of Lemma (49) are used repeatedly. The first is
an L^2 bound for the maximal function of tested adjoint partial sums,
in terms of the tests' l^2 size and the Bessel constant B, with
logarithmic dependence on the catalog budget D and no dependence on the
number of depths. The second is the input statement of (45),
which follows from the two-block structure: any depth-varying schedule
with edge-overlap a and block condition W satisfies (45), and the
nondecreasing restriction to a smaller schedule preserves the same
constant. The third is the statement on increments of inherited
projections, which converts the augmented-catalog construction to a
statement about nested orthogonal projections.
The base reduction converts the schedule F_{j, s} into a structured
tuple Y = B + U with three active patterns, in such a way that the
scalar analysis of ./trilinear-hilbert-l3.pdf may be applied to the
tuples. It proceeds in three steps, analogues of the cyclic
approximation, the high-slot replacement, and the B + U split of
Section 6 there. The tuple Y is a piecewise-fixed approximation of
the raw input in which only a bounded number of active channels
survive. The B part is a bounded-complexity approximation of Y taken
from a bounded active set; the U part collects the residual pieces,
each with a fraction of the total L^2 energy.
Fix an accuracy index t >= 12, put k = k_t, and let eta = e^{-k^theta}
for a fixed 0 < theta < 1. Fix a dyadic root J of interest and a
segment R_0 subset J on which the analysis proceeds. Partition R_0
into attempt roots R_1, R_2, ... with the packing property
sum_{attempt R} |R| <= (4/3) |R_0|, (50)
to be established below. On each attempt root R an approximation
tuple Y = (Y_0, Y_1, Y_2, Y_3) is constructed and analysed independently
of the other attempts.
Cyclic approximation. On every cell I of the dyadic tree the local
form H_I(F_{0, s}, F_{1, s}, F_{2, s}, F_{3, s}) is first expanded
into a two-slot sum. Let g_{e, n, I} denote the last fit in slot e at
or before level n, or zero if no fit exists; and let g_{e, prev, I}
denote the last fit in slot e strictly before the current level. For
u < t with iota(u) = i != j = iota(t), let a, b be the other two
slots and define
X_i^{u,t} = g_{u, I} - g_{u-4, I}, X_j^{u,t} = g_{t, I} - g_{t-4, I},
X_a^{u,t} = F_{a, s} - g_{a, prev, I}, X_b^{u,t} = F_{b, s} - g_{b, prev, I}. (51)
Ordering the slots in their original order,
H_I(F_{., s}) = sum_{u<t, iota(u) != iota(t)} H_I(X^{u,t}_I) + Rem_{n, I}. (52)
The remainder Rem_{n, I} is the term with four residuals
r_{e, n, I} = F_{e, s} - g_{e, n, I} in all four slots, together with,
for each slot e, the term whose e-th entry is the fitted approximant
g_{e, n, I} and whose other entries are the residuals r_{l, n, I}.
The key consequence is that
sum_{I in U} |I| |Rem_{n, I}| -> 0 (53)
as n → ∞ for every fixed finite U subset I.
The increment (51) is the depth-varying analogue of the two-earliest-
increments expansion of Section 6 of ./trilinear-hilbert-l3.pdf,
adapted so that each fitted approximant g_{e, n, I} is a projection of
F_{e, s(I)} at the depth of the cell I. The crucial point, established
in the reference at the single required accuracy, is that every pair
X^{u,t}_I lies inside a bounded-complexity catalog with the birth-cell
and column-size bounds (47). For the depth-varying setting this is
precisely the catalog constructed above in the Bessel-rows analysis:
the fits g_{e, n, I} are built by the stopping argument of Lemma (49),
so their labels persist on descendant cells, and the increments
X^{u,t}_I inherit the same augmented columns.
Delayed high fits. The tuple X = (X_0, X_1, X_2, X_3) assembled from
the two-earliest increments still has a residual structure that is
not uniform across the depth. The delayed high fit replaces the two
higher slots with a better approximation at a fixed delay. In each of
slots j in {2, 3} of the tuple, replace the fit
g_{j, t, I} - g_{j, t - 4, I} (54)
of the raw expansion by the delayed increment
P^{new}_{e, h} F_{e, s} - g_{e, prev, I}, (55)
where P^{new}_{e, h} is a nested sequence of orthogonal projections
built from the augmented catalog at projection depth h = s + K_0.
K_0 is a fixed power of k, chosen (below) large enough to absorb the
absorption constants of the depth-varying high fits.
The construction of P^{new}_{e, h} is carried out in projection-depth
order on each high attempt root R_fit. At each h the algorithm refines
the inherited partition, refits locally, and admits all predetermined
predictors becoming available there. The high family's own extra-fit
catalog begins empty at R_fit and grows only by the stopping argument
of the insertion lemma below. A candidate coarse use cell at this step
has depth s = h - K_0; it is an assigned original request-tree cell
provided h >= K_0. Depths with no candidate still carry the full
refinements and admissions, so no gap in the construction arises.
The replacement from X to Y is measured by
sum_{owners S in S(R_old)} sum_{I owned by S}
|I| |H_I(X_I) - H_I(Y_I)| (56)
over the inherited owners S of the base reduction on R_old. A packing
argument bounds (56) by
D_old^C e_k(D_h) sum_S |S| <= C e^{-k} |R_old|, (57)
where D_old is the catalog budget of the low slots (a fixed power of
k), D_h is the aggregate catalog budget of the high fits (also a
fixed power of k), and
e_k(D_h) = e^{-B} poly(log(2 D_h), 1 + Lambda)
+ (2^{-K_0} + (1 + K_0) sqrt(epsilon)) poly(...). (58)
The choices e^{-B}, 2^{-K_0}, sqrt(epsilon) are each taken small
against the powers of k in D_old^C, at fixed k. Concrete sizes are
chosen in the assembly below.
Decomposition Y = B + U. Once Y is constructed, it is split as
Y = B + U with U supported in the two low slots (say a, b) of Y and B
equal to the projection of Y onto the bounded-complexity active set.
Neither B nor U is required to be a single function; both are piecewise-
constant on the tree of active cells. The scaling is arranged so that
|B| <= 1 pointwise, (59)
sum_{s, G} sum_{I in used at s} |I| ||U_{a, I}||_{2, I}^2 <= Lambda_k^2 |R_old|, (60)
sum_{s, G} sum_{I in used at s} |I| ||U_{b, I}||_{2, I}^2 <= Lambda_k^2 |R_old|.
The bound (59) is the boundedness of the active set; (60) follows from
the projection energy (49) applied to the U-supported rows, with a
factor Lambda_k drawn from the total path energy. In the application
Lambda_k is bounded by a fixed polynomial in k.
The size of the tuple Y at every active cell is under control by
construction: the stopping condition of the cyclic fits ensures that
|H_I(Y_I)| <= eta (61)
on the active cells, while inactive cells contribute at most
e^{-k} |R_old| to the whole-cell sum in aggregate.
The three active patterns and the auxiliary pattern extract the bulk
of the tuple at each active cell. In the two low slots put
z^empty_l = B_l, l = 0, 1, 2, 3,
z^a_a = U_a, z^a_l = B_l for l != a,
z^b_b = U_b, z^b_l = B_l for l != b,
z^{UU}_i = B_i, z^{UU}_j = B_j, z^{UU}_a = U_a, z^{UU}_b = U_b, (62)
where i and j are the indices of the two low slots. The pattern
collection P = {z^empty, z^a, z^b} together with z^{UU} spans the
multilinear expansion
H_I(Y_I) = sum_{z in P} H_I(z_I) + H_I(z^{UU}_I),
H_I(z^{UU}_I) = H_I(B_i, B_j, U_a, U_b). (63)
This is elementary multilinearity of H_I. The complementary pattern
z^{UU} contains both residuals; the three active patterns z^empty, z^a,
z^b each contain at most one residual and are otherwise bounded from
the active set.
Summing over active cells and applying (59)-(61),
sum_{I in A_R} |I| |H_I(Y_I)|
<= sum_{I in A_R} |I| |H_I(z^{UU}_I)|
+ sum_{z in P} sum_{I in A_R} |I| min{eta, |H_I(z_I)|}. (64)
The first summand bounds via Cauchy-Schwarz and (60):
sum_{I in A_R} |I| |H_I(z^{UU}_I)|
<= C eta ( sum_I |I| ||U_{a, I}||_{2, I}^2 )^{1/2}
( sum_I |I| ||U_{b, I}||_{2, I}^2 )^{1/2}
<= C eta Lambda_k^2 |R_old|. (65)
The three pattern sums in (64) are the targets of the localization. Put
M_R(z) = sum_{I in A_R} |I| min{eta, |H_I(z_I)|} (66)
for z in P. Then (64) and (65) combine to give
sum_{I in A_R} |I| |H_I(Y_I)| <= C eta Lambda_k^2 |R_old|
+ sum_{z in P} M_R(z) + e^{-k} |R_old|,
(67)
where the inactive contribution is the O(e^{-k}) absorbed into the
last term. The whole-cell target of the depth-local theorem is now
the sum over R and z of M_R(z), plus the two residual summands.
The reducing sequence so far uses only the augmented catalog, the
Bessel-rows estimate, and elementary multilinearity. The remaining
difficulty sits entirely in bounding M_R(z) for each z in P. The
localization does this by turning the tuple z into a collection of
frozen modes, at which point the scalar analysis of
./trilinear-hilbert-l3.pdf, Sections 6-10, becomes applicable verbatim.
The localization runs over the following objects. A search leaf, a
color history, a pattern z, an attempt root R, and a set A_R of active
cells are fixed. On each cell I of the attempt, the scalar argument of
the reference writes H_I(z_I) as a sum over finitely many groups G,
each group of the form
H_I(z_{0, s}, z_{1, s}, z_{2, s}, z_{3, s})|_{group G}
= H_I(B_{0, s}, B_{1, s}, B_{2, s}, B_{3, s}) coupled with
one residual factor U_{a or b, s} in the appropriate slot, (68)
where the group index G tracks the color history and projection depth.
The number of groups on any spatial path is bounded by a fixed power of
the current catalog budget D_base.
The current-group decomposition turns the scalar amplitude on each
cell into a bounded sum of terms whose individual amplitudes are
governed by one frozen scalar "mode". The construction replicates
Section 6 of ./trilinear-hilbert-l3.pdf in the depth-varying setting;
the sole change is that the input behind every group is F_{l, s}
instead of a single fixed scalar function. Since the input dependence
is exactly the read schedule, the construction's operator identities
go through unchanged while the constants remain those of the reference.
For each leaf and color history there is a colouring of the active
depths into at most L_c = k^{O(1)} collections with pairwise disjoint
interiors. Within one color, the group's output is
Z_{l, G, s} = L_{l, G, s} B'_s F_{l, s}, (69)
where L_{l, G, s} is an individual nested ball projection, the family
{P^j_s} of main projections is nested and inherited along descending
cells, and B'_s = P^m_s ... P^1_s is a shared main projection product.
Every nonzero basic constituent of B'_s has its own specified basic
projection Q^tau as the input-side factor P^1_s; an empty product is
allowed for a shared history.
The absolute value estimates from ./trilinear-hilbert-l3.pdf apply
to Z_{l, G, s} as an operator applied to the scheduled input. Define
the current-group structure on the attempt root R: the sum
H_I(z_I) = sum_G H_I(Z_{l_0, G, s}, z_{l_1, s}, z_{l_2, s}, z_{l_3, s})(I)
+ lower-order terms (70)
splits into a main part G-indexed by the current scalar output in slot
l_0 and the three other bounded basic columns from the active set,
plus lower-order corrections accumulated as part of the e^{-k}
remainder. The number of groups on any path is bounded by D_base^C.
The frozen outputs at the block start u_b have the property that
their coefficients are constant on the whole u_b-cell.
Lemma (Frozen composition with depth inputs). Fix a slot, a search
leaf and its color, and a prospective choice of the offsets of its
groups. For every group G of that color let P_e be finitely many
local mode projections: an orthogonal projection in L^2(J_e; H_mode)
onto constant-coefficient combinations of the restrictions of its
available augmented modes, extended to L^2(R; H_mode) by restricting
the input to J_e and extending the output by zero. Suppose these rows
have joint squared Bessel bound at most G_0 in each group. Then
sum_{G, e} ||P_e R_{l, G} conj(L_{l, G, U_e}) B_{l, U_e} B_{l, U_e} (F_{l, u_b})||_2^2
<= G_0 |R|. (71)
The constant is uniform in the prospective offset choices and in the
number of endpoint cells J_e sharing U_e.
Proof. This is a direct corollary of Lemma (49) and the depth-
varying certified-variation identity of Lemma (Certified variation of
localized B). Fix one
endpoint composition. The source compression identity reduces a
product of a history B and its adjoint applied to a fixed-vector row
to a fixed operator composed with a single read. The main-catalog
range check and the input-support vs. observation depths follow from
the row structure: if the final history factor is nonempty, its adjoint
has final action P_U^1, a main-catalog projection on the whole U; if
the history factor is empty, the range is again a main-catalog
combination, produced by the outer ball projections. In both cases the
composed row reads F_{l, u_b}, the input-support and observation
partitions refine in row order, and Lemma (49) applies with reads
F_{l, u_b} nondecreasing in u_b. Summing over the leaf and color,
using the source bound on the joint operator norm, gives (71). For a
shared history (after the outer ball is removed) the same proof
applies, using the endpoint reads F_{l, u_b} for a fixed block.
The certified-variation identity is the depth-varying analogue of the
source certified-variation lemma.
Lemma (Certified variation of localized B). Fix a source search leaf,
a color history, and either a positive ball endpoint filter or a
shared history. For each individual pad mask admissible at a cell of
the attempt, write the endpoint output Z_{G, s} as a composition of
(i) complementary cumulative filters and their complements (all
contractions), (ii) positive ball averages (contractions), (iii) the
basic projection Q^tau per constituent, (iv) the read F_{l, s}. For
every adjacent pair of attempt-depths r < t and every certified cell
E subset of the attempt,
sum_G ||1_E (Z_{G, t} - Z_{G, r})||_2^2
<= 2 ||1_E (F_{l, t} - F_{l, r})||_2^2
+ 2 G_0 sum_{j} ||1_E (P_{j+1} - P_j) F_{l, r}||_2^2
+ 2 G_0 |E|, (72)
where {P_j} are the nested main projections of the leaf. The constants
G_0 are those of the joint operator estimate (71).
Proof. Fix a pad-mask cell. All factors (i)-(iii) are fixed across s,
so their action is the same operator on any input; their operator
norms multiply to at most G_0 by the source bounds. Insert
B_s = sum_tau eps_tau Q^tau_s F_s at both t and r, and distribute the
difference into
Q^tau_t F_{l, t} - Q^tau_r F_{l, r}
= Q^tau_t (F_{l, t} - F_{l, r}) + (Q^tau_t - Q^tau_r) F_{l, r}. (73)
The first summand multiplied by the fixed per-leaf history is a
contraction applied to F_{l, t} - F_{l, r}; summing over G, e via the
standard bound on the number of rows gives the first term of (72). For
the second summand, insert F_{l, r} at its own depth via the source
certified-usable structure of E (the row's own input-support and
observation partitions both make sense at depth r because r is an
actual use depth of the leaf), and apply the joint Bessel bound of
Lemma (49) with the depth offset t - r. Summing over the rows gives
the second term with the projection increment factors from the source
decomposition
B'_t - B'_r = sum_{j=1}^m P_t^m ... P_t^{j+1}(P_t^j - P_r^j)
P_r^{j-1} ... P_r^1,
where the left factors are contractions and each middle increment is a
Bessel-summed object. The |E| term accounts for the diagonal.
Intervals crossing the start of the attempt contribute zero by the
source convention; their contribution is absorbed into the last G_0 |E|
term via the two endpoint sizes.
The side estimate combines the frozen-output operator bound (71), the
certified-variation identity (72), and the source side analysis of
Section 7 of ./trilinear-hilbert-l3.pdf.
Fix a prospective choice of the group offsets and a source role, say
a B-role with side data
S^{roll}_{l, G, s} = Z_{l, G, s} - Z_{l, G, u_b},
S_{l, G, s} = S^{roll}_{l, G, s} + V_{l, G, b},
V_{l, G, b} = Z_{l, G, b} - Z_{l, G, u_b}, (74)
where Z_{l, G, b} is the exact subtotal of the side modes of the frozen
output at u_b. The B-role identity
(Z_{l, G, s})_phys = (A_{l, G, b})_phys + (S_{l, G, s})_phys
+ E_{l, G, b},
E_{l, G, b} = (Z_{l, G, u_b})_phys - (\tilde Z_{l, G, b})_phys, (75)
splits the total into a core part A_{l, G, b}, the side S_{l, G, s},
and the physical error E_{l, G, b}. In a U-role there is no core, and
one takes S_{l, G, s} = Z^U_{l, G, s} with no physical error.
The certified-variation identity (72) applied to each adjacent pair
gives, with the projections increment bound from the same identity,
sum_s sum_{I in used} |I| ||S^{roll}_{l, G, s}||_{2, I}^2 <= C K G_0 |R|. (76)
The fixed V_{l, G, b} is counted at most K times by the O(K) overlap
of the intervals [u_b, s] for b in the block:
sum_s sum_{I in used} |I| ||V_{l, G, b(s)}||_{2, I}^2 <= K G_0 |R|. (77)
In a U-role, the basic energy argument gives
sum_s sum_{I in used} |I| ||Z^U_{l, G, s}||_{2, I}^2 <= G_0 |R|. (78)
The subadditivity ||S^{roll} + V||^2 <= 2||S^{roll}||^2 + 2||V||^2 gives
the whole S bound (69); the constant K absorption is allowed because
K is a fixed polynomial in k.
The source's componentwise first-crossing construction (Section 7,
Side estimate, of ./trilinear-hilbert-l3.pdf) applies to this finite
list of components. Applied separately to the nonnegative partial path
sums for S^{roll}, for the repeated V, and for the basic U output,
using their respective integrated bounds (76)-(78), it produces
first-crossing cells whose total relative length is at most any
prescribed small fixed quantity. On the retained paths the p-th-power
sums are bounded:
sum_s ||S^{roll}_{l, G, s}||_{2, I_s(x)}^p <= P^C (B-role),
sum_s ||V_{l, G, b(s)}||_{2, I_s(x)}^p <= P^C (B-role),
sum_s ||Z^U_{l, G, s}||_{2, I_s(x)}^p <= P^C (U-role),
sum_s ||S_{l, G, s}||_{2, I_s(x)}^p <= P^C (either role),
(79)
with the exponent p = 1 - eps > 2/3 fixed by the reference, and P the
shorthand for the diagonal threshold parameter used in the source
statement (P = ceil(eta^{-alpha}) for some fixed alpha > 0 chosen
later).
The constants in (76)-(79) depend on the source constants C_5, C_19
through the joint operator norm, and on the block structure of the
attempt via K. The application of the source side estimate requires
only three structural properties: the pad bands A_l^p are nested and
satisfy the pad-sum identity; the transfer identity for two sides
fixes the shared per-band coefficients; and the retained paths satisfy
the per-path caps (79) uniformly. All three hold, with the pad
bands constructed as in Section 7 of ./trilinear-hilbert-l3.pdf and
the caps established by (79). The source near-band cost therefore
applies verbatim:
pure-side contribution on retained paths <= P^C |R|, (80)
and the O(1/P)-weighted main transfer costs at most eta P^C |R|.
The exponents C, C_4 in (80) are those of the source near-band
estimate; the exponents are fixed powers of p, k, and q. Integrating over
the attempt root R and using the certified-variation identity, the
total side contribution on R is at most P^C |R|.
Several auxiliary first-crossing cuts accompany the side estimate.
A mark-failure cut removes cells on which the marker's search has
failed to terminate with the required complexity; the total length of
these cells is negligible in the source construction. A side-energy
first crossing cuts the retained path at the first cell whose
accumulated side energy exceeds its prescribed threshold; the
componentwise structure makes each cut cell at most half the preceding
root. A whole-hull maximal cut removes cells on which the running
maximal function of the two side inputs exceeds sixteen times its root
average; the retained portion of the hull has length at least 7/8 of
its root. A large-complement cut removes cells on which a frozen
complement has exceeded the current complement's norm by more than
C_fr; the flag charged to this cut is at most C_fr^{-2} times the
squared difference size, and the accumulated budget of such cuts is
at most P^C |R|. A powered-freeze cut removes cells whose accumulated
freeze error (accumulated as in the source freeze, large-complement,
and restoration estimates) exceeds the prescribed p-th-power budget;
again the total charged length is at most P^C |R|. Each of these
cuts contributes at most a fixed power of K, P, G_0 to the retained-
cell estimate (stated precisely in Section 7 of
./trilinear-hilbert-l3.pdf) and only removes cells, so subsequent
operations only see shorter trees.
The core estimate handles the high-order (three-core or four-core)
contributions. Its machinery is the "Counts for the tested core tuples"
and "Powered estimate for the core terms" of Sections 8-9 of
./trilinear-hilbert-l3.pdf, applied to the persistent labels and the
side readings supplied by (79).
Representative labels. For a fixed role, group, upper height h, and
type, the "Hereditary representatives" lemma (Section 8 of the
reference) chooses a mode label a in each good PRE predecessor and
inherits it on later contained predecessors. The type is the pair of
floor logarithms, base 1.1, of the predecessor's primary count and
mode count. Write z for the group's primary centre and theta_z for
the horizontal vector in its tag; put vartheta = theta_z / Q_0 for
Q_0 the source's denominator-clearing integer common to the roles of
this group. Let L_h, E_h, Xi be the source shift denominator, shift
range, and separation threshold. For distinct representatives on
comparable actual cells I, I' and any x_* in I cap I',
|omega_a(x_*) - omega_{a'}(x_*) + n . vartheta / L_h|
> (Xi/2) max(|I|^{-1}, |I'|^{-1})
for n in Z^{dim theta_z}, |n|_inf <= E_h. (81)
The identification across depths is through the persistent mode label
and its value omega_a(x_*); two labels whose separation (81) holds
cannot be confounded.
Top rows. For the fixed role, upper height h, type, and dyadic
threshold D_*^{-1} <= sigma <= D_*^2, take for each group G and
representative label a the first actual-use cells on each spatial
branch for which sigma_F >= sigma. Several incomparable tops for the
same group and representative are permitted. Each top row
P_{F, I} is a separated mode row supported on I subseteq U_e, the
u_b-cell of the frozen list; the top itself is a downstream
certificate. Lemma (71) applied to the frozen tops gives their
squared energy bound as G_0 |R|. The first-crossing construction of
the "Top count" Proposition (Section 8 of the reference) gives, on
retained paths,
sum_G |A_j^G| <= min(D_*, P^C sigma_j^{-2}) (82)
uniformly, where A_j^G is the set of distinct representatives used
in the bin [sigma_j, 2 sigma_j) along the retained path for the
fixed role, height, and type. A second consequence of the same
argument is
sigma^2 sum_{tops} |I| <= C (1 + K) G_0 |R|, (83)
obtained by summing the side term of the size comparison first at
one depth and then through a block, using the side energy
sum_b sum_G ||V_{l, G, b}||_2^2 <= K G_0 |R| from (77). The
D_*^{-50} floor in the source estimate is absorbed because
sigma^2 >= D_*^{-2}.
Side incidence. A side role uses the actual same-cell function
S_{l, G, s} in its Fourier windows. Parseval gives its incidence bound
w_s = sigma_l^{-2} ||S_{l, G, s}||_{2, I_s}^2, (84)
whose sum is M_l^G and whose path budget is exactly the p-th-power
sum of (79). Consequently, for the fixed role, height, type, and
threshold, the source "Sparse mass of separated progression tests"
Lemma (Section 9 of the reference) applies verbatim. Its hypotheses —
Hilbert-Weyl test alignment, pure-shift separation, and exclusion of
intermediate ranges by the nonexceptional primary of every successful
component — are consequences of the persistent mode graph and its
parameters. The finite lemma therefore gives, for an absolute
p_0 in (0, 1),
sum_s N_s^G <= L_0 min_{i != j} M_i^G M_j^G, L_0 <= P^C (1 + log(nu/mu))^C,
sum_s (N_s^G)^{p_0} <= P^C (1 + nu + mu^{-1} + Lambda)^C (M')^{2 p_0},
(85)
where N_s^G is the count of successful tuples in a bin, i != j range
over the two side roles, M_i^G and M_j^G are the incidence sums (84),
and nu, mu, Lambda, M' are the source parameters of the reference.
Powered estimate. Combining (82) and (84),
sum_G M_j^G <= P^C sigma_j^{-2} in every role,
with the additional D_* cap in a core role. The reference
interpolates the two inequalities of (85), keeps N_s^G inside its
power, and sums the amplitude and frequency bins. Its "Powered
estimate for the core terms", Equation (37) of Section 9 of the
reference, gives absolute constants eps > 0 and C_4, with
p = 1 - eps > 2/3, and total contribution
core contribution on retained cells <= C_q eta^{eps} P^{C_4} |R|
+ O(e^{-k} |R|). (86)
The original cell minimum min{eta, |H_I(z_I)|} is applied before
subadditivity and the group count is kept inside the power. Numerical
errors are estimated absolutely. The conditional source calculation
therefore uses the supplied persistent labels, full core-subtotal
masks, role counts, and actual side Fourier data, with no further
cross-depth estimate on an original input.
Retained-cell estimate. Combine (67), (80), and (86). The
surviving cells A_R^{ret} of the attempt root satisfy
sum_{I in A_R^{ret}} |I| min{eta, |H_I(z_I)|}
<= sum_{z in P} M_{A_R^{ret}}(z)
<= C_q (eta^{eps} P^{C_4} + eta Lambda_k^2 + P^C) |R|
+ O(e^{-k} |R|). (87)
Because P = ceil(eta^{-alpha}) for some fixed alpha > 0 and
Lambda_k is a fixed polynomial in k, the sum inside the parenthesis
is bounded by C_4 eta^{c_*} for constants c_* > 0 depending on
(alpha, eps, theta), provided k_1 is chosen large enough. The
constants C_4 and c_* depend only on q, c_0, C_0, alpha, eps, theta.
Packing of attempts. The cuts feeding A_R^{ret} are of finitely many
kinds: marking failures, side-energy first crossings, core-energy
first crossings, whole-hull maximal cuts, freeze and large-complement
first crossings, and powered-freeze first crossings. Each cut removes
a set of relative length bounded above uniformly on any attempt tree,
by (87) in every case. The "Packing of attempts" Lemma (Section 3 of
the reference) then gives, for the tree of attempts into which R_0
splits,
sum_{attempt R for segment R_0} |R| <= (4/3) |R_0|. (88)
The geometric factor 4 comes from the 4-ary branching of the segment
tree; the finite recursion on the original finite depth tree
terminates and produces the geometric sum exactly. When a segment R_0
is a disc of the depth tree, the packing applies on the subtree rooted
at R_0 without change.
Choice of q. The fixed-input analysis of ./trilinear-hilbert-l3.pdf
has one structural constant, the reference's parameter q in the L^q
form bound, at which the whole hierarchy of estimates closes. Its
"Order of the parameters" (Section 10 of the reference) requires the
single numerical condition
C_19^ref C_5^ref (q - 2) < 1/8, (89)
where C_19^ref and C_5^ref are the source degree bounds. All other
degree bounds in the reference are fixed before q, and the source
stagger and quality exponents precede q, while its power parameter
and starting accuracy follow q.
The depth-varying analysis acquires its q-dependence through the
absorbed side and core estimates (80) and (86), which together supply
a numerical condition of the same form
C_19 C_5 (q - 2) < 1/8, (90)
with C_19 and C_5 the depth-varying analogues. Both (89) and (90)
are open at q = 2 and the degree bounds are absolute. Choose one
value of q in (2, 3) that satisfies both simultaneously. Use it
throughout the depth-varying analysis and, in Section 10 of the
reference, in the fixed-input closure.
Summation of accuracy layers. With q fixed, the accuracy thresholds
are k_t with k_{t+1} = k_t^A for some A > 1. Then
sum_{t >= 12} t e^{-c_* k_t^theta} < ∞, (91)
and the retained-cell estimate (87) accumulates over t as
sum_{I in A_J^{ret}} |I| min{eta_t, |H_I(z_I)|}
<= C_q sum_{t >= 12} ( eta_t^{c_*} + e^{-c_* k_t^theta} ) |J|
+ O(e^{-k_1}) |J|. (92)
The finitely many initial layers (t < 12) each absorb into the
constant C_loc(c_0, C_0) with an additive factor depending only on
c_0, C_0, q.
Sixteen lines and one inequality close the depth-local theorem. The
scheduled input F_{j, s} is bounded on every descendant by (44) and
on every block by (45). The local forms are the same admissible
H_I of (17). The depth-local theorem's conclusion (46) follows from
(67), (80), (86), (88), (92) combined: each attempt root R
contributes at most C_q eta^{c_*} |R| + O(e^{-k_1})|R|, the sum
over attempts is at most (4/3) |R_0|, and the initial finitely many
layers add a bounded constant. The resulting bound is
sum_{I in tree} |I| |H_I(F_{., s(I)})|
<= C_loc(c_0, C_0) |J|, (93)
with C_loc depending only on c_0, C_0 through the fixed constants
of the reference.
The orbit oscillation estimate (43) follows by substituting (93)
into (27) of the continuous test from translation averaging. Tracing
the reductions one more time:
(93) ==> (27) ==> (25) continuous test
==> (29) integer sequence test
==> (39) discrete oscillation bound
==> (43) orbit oscillation estimate.
The oscillation method (7) applied to Z_k = V_k^{phi} then produces
the smoothed averages' almost everywhere convergence with a limit
identified as p = prod_j int f_j d mu by (9) and Fatou. The grid
comparison (12) and its interpolation paragraph produce
lim_{N -> infinity} A_N(x) = p (94)
for mu-almost every x. Since p = prod_j int f_j d mu, the main
theorem follows.
The constants in every step are numerical and independent of the
probability space, of the underlying transformation, of the number of
depths, and of the number of blocks in the schedule. Each step obeys
one uniformity condition: mixing is uniform in |m| -> infinity for
functions bounded in L^infinity; van der Corput's lemma is uniform
in the norm bound M; the Bessel rows lemma is uniform in the catalog
budget D and the input schedule; the localization and side-core
estimates are uniform in the number of groups, depths, and blocks by
the source packing; and the summation over accuracy layers is
uniform in the depth range because the finitely many initial layers
absorb into the constant C_loc.