After the successful height: from scalar Fourier expansions to frequency tests

This chapter gives a close reading of the fixed October 5 version of An $L^3$ bound for the trilinear Hilbert transform, specifically equation (29) on page 78 and equations (30)–(33) in Section 9.2 on pages 79–81.

The aim is to explain how the finite mode expansion at a successful height becomes a frequency test with decaying weights, and why, with three core roles, the fourth role must carry a squared Fourier mass. Finite sparse counting is treated by Lemma 9.1 in the following Section 9.3, not Section 9.1. That counting lemma is not reproved here.

Back to the guide to H

1. The assumptions from which the chapter starts

Fix the root $R$ of a retained attempt, a localization leaf, an actual-use cell $I=I_s$, and a group $G$. Write

All mode lists are finite. The set of core roles $J\subset\{0,1,2,3\}$ has size $m_0=3$ or $4$. If there are three core roles, denote the remaining side role by $l$. We continue to use $c=(-1,3,-3,1)$.

The inputs are uniqueness of the successful predecessor and constancy of predicates on minimal CURRENT components from Lemma 8.4; inherited representatives and separation from Lemma 8.10; component almost orthogonality from Lemma 8.11; the component sizes in (26); and the path bounds of Proposition 8.12. The Fourier expansion below does not by itself establish this structure.

The parameter $D_*$ controls the finite catalog and the absolute coefficient bounds, with $\log D_*\le\operatorname{poly}(P,k)$. All precision exponents are fixed before the pre-lag and Fourier truncation are chosen. Raising $D_*$ to a fixed power preserves its logarithmic complexity bound. It is the depth of the actual cell that belongs to the depth range of the relevant block; the spatial interval $I$ is not itself a depth interval.

2. The scalar model at a successful height

First specify the parameters. Let $z$ be the common curvature tag of the group, let $Q_0$ be a common integer clearing the denominators of $C_z$, and set $\vartheta=\theta_z/Q_0$. Section 8.3 of the source fixes $\Delta=10$ and takes

Here $A_0$ is a sufficiently large integer chosen in terms of the preceding height, Fourier truncation, and catalog bounds, while $J_*$ is the finite maximum height. The quantity $E_h$ is an integer shift radius, not a projection operator. Each height is drawn independently and uniformly from the upper range specified in the source,

with $J_*$ previously chosen large enough that its lower endpoint is at least $\Delta$. Predicate constancy on minimal CURRENT components and the bad-density bound $2\delta_*$ for each core mode are both used on this range. Fix the final successful height $h\in\mathcal H_{\mathrm{up}}$ and put

Here $a_j$ is the inherited representative of the chosen predecessor. The PRE path and the pre-lag express the slope of every selected core mode as $\gamma_j^*+n_j\cdot\xi$, up to an error that is tiny after multiplication by $r$. The quadratic Taylor error is tiny as well.

The physical core input at role $j$ therefore satisfies (29) in the source:

Let $A_j$ be the sum of the absolute selected mode coefficients. We can arrange

We first use the actual periodic cutoff multiplied by the finite trigonometric polynomial of modes, identifying the common progression multiplier. Only afterwards is the cutoff replaced by a finite Fourier truncation. The actual selected input and the actual cutoff model both have physical absolute value at most $A_j$; the truncated model can be bounded by $2A_j$.

Including the multiplier, all required Fourier supports, in coordinates anchored at any role, lie in the full integer box

The box must include the ranges of every convolution and every later difference. After an allowed enlargement of the bounds, $D_*A_j$ bounds the Wiener norm of each $g_j$, and $D_*$ bounds the Wiener norm of the truncated multiplier. The precision choices also put the truncation tail below the required fixed inverse-power error.

3. The tiny/huge condition applies to frequency differences

A successful height supplies a nonexceptional primary and a rational subspace $K_B\subset\mathbb R^d$, where $d$ is the current horizontal dimension. We need the following quantitative dichotomy for every character difference $n$ in the full Fourier range:

The small bound can be any prescribed inverse power of $D_*$, and the large bound can be any prescribed fixed power. Below we require $M_{\mathrm{gap}}>10D_*^{250}$, with $\varepsilon_{\mathrm{gap}}$ small enough to absorb all finite Wiener-norm costs.

Testing only the original frequencies is insufficient. Two large frequencies may have a small difference. Both the squared norm and separation of reference frequencies require the dichotomy for their differences.

Set

If a nonzero integer $q$ satisfies $qn\in\mathcal L$, then $n\in K_B$ and hence $n\in\mathcal L$. Thus $\mathcal L$ is a saturated lattice. An integer change of basis identifies it with a coordinate sublattice having torsion-free quotient, so its annihilator $H$ is a connected torus, including the zero-dimensional case. Its character group is $\mathbb Z^d/\mathcal L$, and

All Haar measures are normalized to be probability measures.

4. Compare interval square averages with Haar square averages

Write the finite expansion $g(u)=\sum_n\widehat g(n)e(n\cdot u)$. For a difference frequency $v$, the normalized interval average is

If $v\in\mathcal L$, then $|y-x^*|\le r$ gives

If $v\notin\mathcal L$, direct integration of the exponential gives

Expand $|g|^2$ and compare each term with its Haar character integral:

All sums are finite, so no uncontrolled interchange of infinite sums is involved. After the precision choices, the right-hand side is at most $D_*^{-100}A_j^2$.

We now explain the component bound in (30). Let $\mathcal D_j^{\mathrm{sel}}$ denote the selected minimal CURRENT components. Predicate constancy means that an entire core subtotal $A_D$ is selected at once. Define

Equation (26) in the source is $\sigma_{F_j}^2=\sum_{D\subset F_j}w_D$. For arbitrary coefficient subtotals in distinct minimal components, Lemma 8.11 gives

Only this upper bound is needed. The model error and the square root of the comparison-of-averages error therefore give

Squaring yields

To check absorption of the error, if there are at most $D_*$ components, then

The last term is already included in $\sum_Dw_D$. The enhanced-space norm bounds the physical component, so dropping the private coordinates does not enlarge the bound needed on the right.

5. What the four-factor Haar estimate uses

On $H^2$, consider the four progression coordinates $u+(j-l')v$. Any two distinct coordinates define an integer linear map with nonzero integer determinant $j'-j$. Multiplication by a nonzero integer is surjective on a connected torus, so these two coordinates have independent Haar distributions.

The particular four-factor bound needed here also has a direct proof:

The two adjacent-pair maps both have determinant $1$, so this particular $L^2$ bound holds on every compact abelian group. Connectedness is used for the stronger statement above about independence of any two coordinates; it is not a necessary condition for the adjacent-pair bound. Inserting a uniformly bounded multiplier only contributes its supremum norm.

6. Four lines on which the Fourier transform vanishes

Anchor the coordinates at role $l'$ and set

Denote the normalized kernel by $W_s(h,v)$, and define the Fourier transform with a positive sign:

All four marginal integrals of the original kernel vanish. Fix $j$ and change variables by $u=h+(j-l')v$. Then

For a general anchor, the four slopes are therefore $j-l'$. The four-core argument below takes $l'=0$; only with that choice can the lines be written as $\beta=j\alpha$.

Smooth compact support gives rapid decay of $\mathcal F_s$ and all its derivatives, with constants uniform in the cell $I$. Write

When $d\le1$, apply the mean value theorem towards the nearest vanishing line; the frequency size at nearby points is comparable to that at the original point. When $d>1$, use rapid decay directly. Thus, for every fixed $N$,

Zero distance is handled by vanishing before any positive dyadic scale is assigned.

7. Four cores: retain one full pair of cosets

Put $U_0=D_*^{250}$ and define

Let

Here $R_z$ is the truncated smooth periodic multiplier arising from the actual common branch phase, with a uniform bound. Its Fourier expansion writes the local form as

Search the full box for a pair $(n_0,m_0)$ whose two normalized real frequencies both have absolute value at most $U_0$. The reference is chosen from the full box using the cell, the representatives, and a fixed rule; it is independent of the historical failure mask.

If two frequency pairs are both moderate, their difference in each coordinate is at most $2U_0$. Gap forces the corresponding integer differences to lie in $\mathcal L$. Thus moderate frequencies belong to at most one pair of cosets,

Within one coset, the normalized real frequency differs from the reference by at most $\varepsilon_{\mathrm{gap}}$. The unnormalized difference is at most $\varepsilon_{\mathrm{gap}}/r$.

Retain every actual Fourier coefficient in this pair of cosets, even if a frequency has crossed the $U_0$ boundary by a tiny error. The full coefficient sum is exactly

Keeping only the portion below the threshold would destroy this Haar coefficient identity.

8. The four-core bound and its errors

The four-factor estimate bounds the absolute value of the full-coset coefficient by $C\prod_ja_j'$. Within the retained coset, replace the Fourier kernel by its reference value. Its uniform Lipschitz bound and the Wiener-norm bounds make the resulting absolute error sufficiently small. Every other coset has no moderate point, so rapid decay applies.

Let $(\alpha_0,\beta_0)$ be the normalized reference frequencies. If there is no reference, or if

then the entire term can be included in the absolute error. Otherwise, assign dyadic parameters to the positive quantities:

The four-core contribution is therefore bounded by

plus the allowed absolute error. When discarding terms near a vanishing line, first use the absolute coefficient sum and $U_0^{-1}$. Do not first raise a small error to a power below $1$.

9. Three cores: first remove the common branch phase from the side role

Let $l$ be the side role, and insert a freshly translated random cutoff $\zeta^{\rho_l}$. Its expectation is the spatially constant quantity

Keep the factor $c_\zeta^{-1}$ outside the averaging of the original form, and define

Since $|\zeta^{\rho_l}|\le1$ and the branch phase has modulus $1$,

The minus sign is necessary: the common branch phase in the source model must be completed with the side role before it can be expressed as a periodic multiplier.

Now use $l'=l$ as the anchor, and set

Search only the vertical box for a reference $m_0$ satisfying

If there is no reference, or if $|\beta|<U_0^{-1}$, include the contribution in the absolute error. Otherwise, define

Here we use only the vanishing line $\beta=0$ corresponding to the anchor role.

10. Vertical integration leaves an $L^2$ sequence of horizontal coefficients

Define

This is supported in the interior of the anchor interval. Vanishing of the anchor marginal gives $W_0(h)=0$. Use the mean value theorem when $\beta$ is small, and integrate by parts in $v$ at large frequencies. Thus every fixed-order derivative in $h$ satisfies

Let

For any $\|q\|_{L^2(H)}\le1$, insert $\overline q(u)$ as the fourth factor at the anchor. The four-factor estimate gives

Hilbert-space duality implies

Group its Fourier coefficients by $\mathbb Z^d/\mathcal L$. Parseval bounds the $\ell^2$ norm of the grouped coefficient sequence by the same quantity. The relevant core size is this norm, rather than the absolute sum of the coefficients.

Choose one reference $n$ from the full box for each horizontal coset. The normalized real frequencies $rn\cdot\xi$ of distinct references differ by more than $1$, while normalized real frequencies in the same coset differ by at most $\varepsilon_{\mathrm{gap}}$. When replacing other frequencies by the reference, bound the error through an $L^2$ pairing with the side input. The side input need not have absolutely summable Fourier coefficients.

11. The Schur bound requires both row and column counts

Expand the side input in the orthogonal Fourier basis on $I$:

Begin with a finite truncation of the side Fourier series. The Schur bound below is independent of the truncation length. Once it is proved, convergence of the Fourier series in $L^2(I)$ and continuity of pairing with the fixed bounded smooth kernel allow the truncation length to tend to infinity. Neither pointwise nor absolute convergence is required.

For a horizontal reference $n$, write

The entries of the remaining matrix pairing are controlled Fourier integrals, giving

The sum in this expression combines a normalized horizontal real frequency with an integer side Fourier frequency.

Decompose into shells of dyadic radius $\Lambda=1,2,4,\ldots$. In a fixed row $n$, integrality of $k'$ gives at most $C\Lambda$ entries in the shell. In a fixed column $k'$, the frequencies $\lambda_n$ are pairwise $1$-separated, again giving at most $C\Lambda$ entries. Thus the maximum absolute row sum and column sum of the shell matrix are both at most

Two applications of weighted Cauchy–Schwarz give Schur's inequality directly:

Taking $A\ge N+1$ gives $C_N\mu\nu^{-N}\Lambda^{-N}$. Counting rows through the integer frequencies, without the column count supplied by separation of the horizontal references, would not prove this bound.

12. Full windows and averaging over a fresh offset

Define the window

This uses the full box, not merely the modes selected by the current failure mask. Define the square root of the averaged squared Fourier mass by

For each fresh offset, first apply the matrix bound and the horizontal coefficient norm estimate. Then use Cauchy–Schwarz in the offset to obtain the form of (32) in the source:

The external factor $c_\zeta^{-1}\le2$ has been absorbed into the absolute constant. Shells larger than $U_0$ contribute only to the absolute error, by the same Schur estimate or an ordinary $L^2$ pairing.

For a single window, Parseval only gives $b_\Lambda^2\le\|S\|_{2,I}^2$. Summing squares over many triples requires disjointness or bounded overlap of the corresponding windows. For example, if different sets of window centres are separated by more than $4\Lambda$, their windows of radius $2\Lambda$ are disjoint. Later separation tests establish this geometric condition. Parseval for one window does not by itself bound the sum over arbitrary triples.

All tested triples in the same cell must use the same fresh offset $\rho_l$. This allows summation over disjoint windows for each fixed offset, followed by expectation. Resampling the offset separately for every triple would lose this direct common square bound.

13. Why the absolute error does not count failure histories

Replacing models role by role, Fourier truncation, frequency replacement within a coset, far-frequency tails, and discarding terms near vanishing lines can all be bounded by

with an additional factor $\|S\|_{2,I}$ when there is a side role. For example, multiplying the model error $D_*^{-150}A_j$ by the physical bounds for the other roles leaves ample room. Even after the finite Wiener-norm costs, the error exponent remains substantially better than $-100$. All precision powers are fixed before the parameters are chosen.

The key is to expand the absolute coefficient sum by pure mode tuples first. Draw each height independently and uniformly from $\mathcal H_{\mathrm{up}}$, and retain the first draw at which every role in the tuple succeeds. The success probability of one draw is at least

so the total probability of all first-success histories for this tuple is exactly $1$. Averaging the error over all histories therefore leaves just the absolute coefficient product for that pure tuple, with no factor counting failure histories.

To integrate these errors along paths to obtain $O(e^{-k}|R|)$, the source also uses path bounds on the catalog, coefficients, and retained sizes. This chapter proves the local absolute error and the absence of a history-count loss. The global path bounds remain inputs from Section 8. Numerical errors are always estimated separately before taking a power below $1$.

14. From failure counts to probability powers in the component sizes

Fix the final successful height and the component tuple. If there were $n$ preceding failures, assign each failure to the first role that failed. There are at most $m_0^n$ such assignments. If role $j$ receives $n_j$ failures, then $\sum_jn_j=n$.

Predicates are constant on minimal CURRENT components, so selection acts as an indicator on the entire weight $w_D$. At each independent height assigned to this role, the failure probability is at most $2\delta_*$. Dropping the additional success requirements can only increase the probability. The component estimate therefore gives

The expectation is taken with respect to the original product probability of the previously sampled heights, with the failure and success requirements retained in the mask indicators. It is not conditional on the first $n$ draws having already failed; that conditioning would remove the small probability factors.

The roles share the same heights and are not independent. Apply Hölder, then use the deterministic upper bound to pass from second moments to moments of order $m_0$:

All histories are therefore controlled by the geometric series

provided we choose $m_0(2\delta_*)^{1/m_0}<1$ in advance.

15. What equation (33) supplies to the next section

Finally, sort by final height, type, and dyadic size. For a core role, $\sigma_j$ is the size class from (26); for the side role, $\sigma_l$ is the size class of $b_\Lambda$. In the four-core case, set $\Lambda=1$. Once all frequency and size parameters are fixed, the local contribution is bounded by

plus the absolute error treated separately above. The quantity $N_s^G$ counts core-label tuples that pass the non-negligible frequency test at the current cell. A given tuple is counted at most once per scale.

This is the finite combinatorial input to Lemma 9.1 in Section 9.3. The later power estimates must keep $N_s^G$ inside the power and use the minimum on the original cell before applying subadditivity. The three-core Fourier bound alone does not prove the full sparse count across scales, nor does this chapter establish the main theorem of H.