Detecting return increments with two observation coordinates

This chapter treats the analytic argument for distinct slots in Lemma 6.1 of H, on pages 40–41 of the fixed October 5 version of An $L^3$ bound for the trilinear Hilbert transform. The phase calculation for returns comes from the proof of Lemma 4.3 on pages 16–17.

The goal is specific: a non-negligible quadrilinear correlation forces a uniform bound on the curvature frequency increments of every sufficiently small return. Large real frequencies remain in the calculation. This chapter stops before the construction of a rational fast space, which requires a further argument from the geometry of numbers.

Back to the guide to H

1. Which data have complexity bounds?

Put

e(t)=\exp(2\pi\mathrm{i}t),\quad(c_0,c_1,c_2,c_3)=(-1,3,-3,1).

The two source slots are $i,j$ , and the other two, $a,b$ , are the observation slots. Together the four indices are exactly $\{0,1,2,3\}$ . The source columns have the form

\phi_l(y)=e(c_l\lambda_l y^2+\beta_l y) \sum_{n\in\mathbb Z^{d_l}}\chi_l(\theta_l y-n) e\!\left(c_l(\theta_l y)^TC_l(\theta_l y-n)\right), \quad l=i,j. \tag{Columns}

Following Definition 4.1 on page 15 of H, fix $H\ge \mathrm{e}^2$ and suppose that both source atoms have complexity at most $H$ . More explicitly,

d_l\le\log H,\quad \operatorname{ht}(C_l)\le H,\quad \operatorname{supp}\chi_l\subset[-H,H]^{d_l},\quad \|\chi_l\|_\infty+\max_\nu\|\partial_\nu\chi_l\|_\infty\le H^{C_0},

where $C_0$ is a fixed absolute constant. Higher Gevrey derivative bounds are part of the source definition, but this chapter uses only the first derivative bound. Thus the total dimension satisfies $d_i+d_j\le2\log H$ . The parameters $\theta_l,\lambda_l,\beta_l$ can be arbitrary real numbers, with no restriction on their size.

The physical columns have a fixed bound, which we write as $|\phi_l|\le1$ to simplify constants. Any other controlled bound can be absorbed into $D$ below. Every occurrence of $D$ can be chosen quasipolynomial in the original complexity; in H this means $\exp(k^{O(1)})$ .

2. Use the two observation coordinates

The four coordinates in the original quadrilinear integral are $y_l=x+lt$ . Let the observation interval have centre $x_I$ and length $r$ , and define

y_a=x_I+rz,\quad y_b=x_I+rw.

Solving for the original variables gives

x=x_I+r\frac{bz-aw}{b-a},\quad t=r\frac{w-z}{b-a},\quad \left|\frac{\partial(x,t)}{\partial(z,w)}\right|=\frac{r^2}{|b-a|}.

At each slot,

Y_l=x_I+r(\xi_lz+\zeta_lw),\quad \xi_l=\frac{b-l}{b-a},\quad \zeta_l=\frac{l-a}{b-a}.

The four source-slot coefficients $\xi_i,\zeta_i,\xi_j,\zeta_j$ are nonzero. The identities

\sum_{l=0}^3c_l l^u=0,\quad u=0,1,2

give

c_i\xi_i\zeta_i=-c_j\xi_j\zeta_j=\kappa\ne0. \tag{Mixed coefficient}

This can also be checked by substituting the four distinct integers directly. The denominator is at most $9$ , and the nonzero numerator is an integer, so $|\kappa|\ge1/9$ .

Absorb the scale kernel, the smooth localization, and the fixed Jacobian into $W(z,w)$ , a bounded Lipschitz function on a fixed box. Extend the observation functions by zero outside the box. After normalization,

\|f\|_2,\|g\|_2\le1,\quad \operatorname{supp}f,\operatorname{supp}g\subset[-1/2,1/2].

Define a single fixed kernel

K(z,w)=W(z,w)\phi_i(Y_i)\phi_j(Y_j).

We may assume that $K$ is supported in a fixed box and that $\|K\|_\infty+\|K\|_2\le D$ . The original correlation reduces to

\mathcal C=\iint K(z,w)f(z)g(w)\,\mathrm{d}z\,\mathrm{d}w, \quad |\mathcal C|>\tau,\quad 0<\tau\le1. \tag{Correlation}

Controlled fixed factors only change $\tau$ by a controlled constant factor. Every kernel below refers to this same $K$ ; it is not redefined for each multiple of a return.

3. Returns and the integer that clears denominators

Set

V_l=r\theta_l,\quad V=(V_i,V_j),\quad C=\operatorname{diag}(C_i,-C_j),\quad A=C-C^T,\quad \Lambda=r^2(\lambda_i-\lambda_j).

Let $q_C$ be the product of the reduced denominators of all entries of the two matrices. Then $q_CC_i,q_CC_j$ are integer matrices, and

q_C\le H^{d_i^2+d_j^2},\quad \log(6q_C)\le C(1+\log H)^3.

Take

P_0=6q_C.

The denominator of $\xi_l$ divides one of $1,2,3$ . Thus $m_l\in P_0\mathbb Z^{d_l}$ implies $\xi_lm_l\in q_C\mathbb Z^{d_l}$ .

A return consists of

h\in\mathbb R,\quad m=(m_i,m_j)\in P_0\mathbb Z^{d_i+d_j}, \quad d_h=Vh-m

Its smallness is measured by $|h|+\|d_h\|_2$ , not by $\|m\|_2$ . Define

\Omega(h,m)=(2\Lambda+V^TCV)h-V^TAm. \tag{Increment}

We will give an explicit $\varepsilon_{\mathrm{ret}}>0$ such that

|h|+\|d_h\|_2\le\varepsilon_{\mathrm{ret}} \quad\Longrightarrow\quad |\Omega(h,m)|<9. \tag{Target}

The reciprocal of this threshold is quasipolynomial in the complexity and $\tau^{-1}$ , and is independent of $|\theta_l|,|\lambda_l|,|\beta_l|$ .

4. The full translation calculation for one source phase

Translate only the first observation coordinate, $z\mapsto z+h$ . Fix a source slot $l$ and temporarily suppress its index. Write

u=\theta Y,\quad q=u-n,\quad m'=\xi m,\quad \delta=\xi(Vh-m).

After translation, $u$ becomes $u+m'+\delta$ . Write the new summation index as $n+m'$ ; this is a valid change of integer index. The new horizontal remainder is $q+\delta$ .

After removing the fixed coefficient $c_l$ , the change in the horizontal quadratic phase is exactly

\begin{aligned} &(u+m'+\delta)^TC(q+\delta)-u^TCq\\ &\quad=u^TC\delta+m'^TCu+\delta^TCq +(m'+\delta)^TC\delta-m'^TCn. \end{aligned} \tag{Expansion}

The last term is an integer because $m'\in q_C\mathbb Z^d$ . The first two terms can be rewritten as

u^TC\delta+m'^TCu =\xi u^T(CVh-Am). \tag{Rearrangement}

The quantity $(m'+\delta)^TC\delta$ may be large. Since it is independent of $z,w$ , it must be included in the constant phase. Smallness of $\delta$ alone does not make this term a small error.

The only remaining variable phase error is $\delta^TCq$ . On the support of the amplitude, $q$ is bounded in terms of the complexity, so the absolute value of this error is at most a controlled constant times $\|\delta\|_2$ . The amplitude change

\chi(q+\delta)-\chi(q)

is likewise controlled by the first derivative bound. On the union of the original support and a sufficiently small translate, at most $(2H+5)^{d_l}$ integer indices contribute at any point; the logarithm of this number is $O((1+\log H)^2)$ . The matrix norms and the first derivative bounds satisfy the same type of bound. Summing over these indices therefore still gives an error $D\|d_h\|_2$ , where $D\le\exp(C(1+\log H)^C)$ . No bound on $u$ or $m'$ is needed.

5. Keep all large parameters in the exact phase

Restore the indices and define

A_l=C_l-C_l^T,\quad \Omega_l=(2r^2\lambda_l+V_l^TC_lV_l)h-V_l^TA_lm_l.

Combining the ordinary quadratic phase with the variable linear term from Section 4 gives

\phi_l(Y_l(z+h,w)) =e(\gamma_l)e(c_l\xi_l^2\Omega_lz+c_l\xi_l\zeta_l\Omega_lw) \phi_l(Y_l(z,w))+E_l(z,w),

where $\|E_l\|_\infty\le D\|d_h\|_2$ . To account explicitly for the constant terms, we can take

\begin{aligned} \gamma_l={}&c_l\lambda_l\!\left(2x_Ir\xi_lh+(r\xi_lh)^2\right) +\beta_lr\xi_lh\\ &+c_l\xi_l(\theta_lx_I)^T(C_lV_lh-A_lm_l) +c_l(m_l'+\delta_l)^TC_l\delta_l,\\ m_l'={}&\xi_lm_l,\quad \delta_l=\xi_l(V_lh-m_l). \end{aligned}

This accounts for every potentially large term involving $\theta_lx_I,\lambda_l,\beta_l$ or $m_l'\delta_l$ .

By the definitions of the difference data,

\Omega_i-\Omega_j=\Omega(h,m).

so the combined $w$ frequency of the two sources is exactly $\kappa\Omega(h,m)$ . Set

\mathcal A=c_i\xi_i^2\Omega_i+c_j\xi_j^2\Omega_j,\quad \gamma=\gamma_i+\gamma_j.

Translating the Lipschitz kernel $W$ contributes a further error $D|h|$ . Increasing $D\ge1$ as necessary, whenever $|h|+\|d_h\|_2\le1/3$ we have

K(z+h,w)=e(\gamma)e(\mathcal Az+\kappa\Omega w)K(z,w)+E(z,w), \quad \|E\|_\infty\le D(|h|+\|d_h\|_2). \tag{Kernel translation}

This identity holds on the fixed box used in the correlation integral. The translated truncated supports remain inside a fixed, slightly larger box.

6. Multiples of a return use the same kernel

For an integer $\ell\ge1$ , the return $(\ell h,\ell m)$ has horizontal error $\ell d_h$ . Both the frequency increment and the $z$ frequency scale linearly:

\Omega(\ell h,\ell m)=\ell\Omega(h,m),\quad \mathcal A(\ell h,\ell m)=\ell\mathcal A(h,m).

The constant phase $\gamma_\ell$ may contain $\ell^2$ terms, but $e(\gamma_\ell)$ always has modulus $1$ . No linear dependence is needed for $\gamma_\ell$ .

When $\ell(|h|+\|d_h\|_2)\le1/3$ ,

K(z+\ell h,w) =e(\gamma_\ell)e(\ell\mathcal Az+\ell\kappa\Omega w)K(z,w)+E_\ell(z,w), \quad \|E_\ell\|_\infty\le D\ell(|h|+\|d_h\|_2). \tag{Multiples}

The kernel on the right is always the original $K$ . Only the external modulations, constants, and errors change.

7. Truncate large values while retaining the correlation

Take

B=1+\frac{16D}{\tau},\quad f_B=f\,\mathbf1_{\{|f|\le B\}},\quad g_B=g\,\mathbf1_{\{|g|\le B\}}.

The $L^2$ normalization gives

\|f-f_B\|_1\le B^{-1},\quad \|g-g_B\|_1\le B^{-1},

and $\|f\|_1,\|g\|_1,\|f_B\|_1,\|g_B\|_1\le1$ . Hence

\left|\mathcal C-\iint Kf_Bg_B\right| \le\frac{2D}{B}<\frac{\tau}{8}. \tag{Truncation error}

The fixed truncated correlation therefore has absolute value greater than $7\tau/8$ . Meanwhile, $f_B,g_B$ still have $L^2$ norms at most $1$ and absolute values at most $B$ .

This follows the truncation argument in the source. Truncation is not a minimal requirement when the kernel is fixed and bounded: the later estimates can also use the kernel bound more directly. What one cannot assume is that an arbitrary $L^2$ multiplier preserves a Bessel bound for exponentials with a constant independent of the multiplier.

8. Derive the Bessel bound on a fixed interval

Suppose that $F\in L^2(\mathbb R)$ is supported in an interval of length at most $1$ , and that $|\alpha|\ge1$ . Put $p=|\alpha|^{-1}\le1$ and periodize on $[0,p)$ :

F_p(w)=\sum_{k\in\mathbb Z}F(w+kp).

At each point there are at most $p^{-1}+2$ nonzero terms, so

\int_0^p|F_p(w)|^2\,\mathrm{d}w \le(p^{-1}+2)\|F\|_2^2.

Parseval's identity with period $p$ gives

\begin{aligned} \sum_{\ell\in\mathbb Z} \left|\int_{\mathbb R}F(w)e(\ell\alpha w)\,\mathrm{d}w\right|^2 &=p\int_0^p|F_p(w)|^2\,\mathrm{d}w\\ &\le3\|F\|_2^2. \end{aligned} \tag{Bessel}

Changing the sign only permutes the integer indices and does not affect the identity. The calculation can first be made for integrable square-integrable functions and then extended by $L^2$ approximation. The finite interval ensures that all the integrals are defined.

For almost every $z$ , take $F_z(w)=K(z,w)g_B(w)$ . Its support in $w$ has length at most $1$ . If $|\kappa\Omega|\ge1$ , Tonelli's theorem and the preceding bound give

\sum_{\ell=1}^M\int \left|\int K(z,w)g_B(w)e(\ell\kappa\Omega w)\,\mathrm{d}w\right|^2 \mathrm{d}z \le3B^2\|K\|_2^2\le3B^2D^2. \tag{Fixed-kernel bound}

The first observation function is not assumed to be the same for different $\ell$ .

9. Choose the number of multiples before the return threshold

Now define, in this order,

M=\left\lceil100D^2B^2\tau^{-2}\right\rceil,\quad \varepsilon_{\mathrm{ret}}=\frac{\tau}{1000DM}. \tag{Parameter order}

Since $D\ge1,\tau\le1$ , we have $M\varepsilon_{\mathrm{ret}}\le1/1000$ . Thus every return $(h,m)$ below the threshold has its first $M$ multiples within the range of Section 6.

In the same truncated correlation, change variables by $z\mapsto z+\ell h$ and substitute the kernel translation formula. Define

f_\ell(z)=f_B(z+\ell h)e(\ell\mathcal Az),\quad g_\ell(w)=g_B(w)e(\ell\kappa\Omega w).

Translations and modulations of modulus $1$ preserve the $L^2$ norm, so $\|f_\ell\|_2\le1$ . The error satisfies

\left|\iint E_\ell(z,w)f_B(z+\ell h)g_B(w)\,\mathrm{d}z\,\mathrm{d}w\right| \le D\ell\varepsilon_{\mathrm{ret}} \le\tau/1000.

Consequently, for every $1\le\ell\le M$ ,

\left|\iint K(z,w)f_\ell(z)g_\ell(w)\,\mathrm{d}z\,\mathrm{d}w\right|>\tau/2. \tag{Repeated correlation}

The parameter order ensures that the error bound holds for the entire finite family at once. One cannot first fix the threshold and then increase $M$ without restriction.

10. The energy of the fixed kernel rules out these correlations

Suppose that $|\kappa\Omega|\ge1$ . Apply Cauchy–Schwarz only in the $z$ variable, then sum:

\begin{aligned} &\sum_{\ell=1}^M \left|\iint K(z,w)f_\ell(z)g_\ell(w)\,\mathrm{d}z\,\mathrm{d}w\right|^2\\ &\quad\le\sum_{\ell=1}^M\int \left|\int K(z,w)g_B(w)e(\ell\kappa\Omega w)\,\mathrm{d}w\right|^2 \mathrm{d}z \le3B^2D^2. \end{aligned}

On the other hand, the repeated correlations give the strict lower bound $M\tau^2/4\ge25D^2B^2$ , a contradiction. Therefore

|\kappa\Omega(h,m)|<1,\quad |\Omega(h,m)|<|\kappa|^{-1}\le9

for every return satisfying the same threshold. The parameters $D,B,M,\varepsilon_{\mathrm{ret}}$ are all chosen in advance from the complexity and $\tau$ , independently of the return being tested.

11. The exact quantifiers and the next step

The conclusion is the following. With the source columns, scale, smooth kernel, and normalization fixed as in this chapter, a correlation exceeding $\tau$ yields a controlled $\varepsilon_{\mathrm{ret}}>0$ such that every

h\in\mathbb R,\quad m\in P_0\mathbb Z^{d_i+d_j},\quad |h|+\|Vh-m\|_2\le\varepsilon_{\mathrm{ret}}

satisfies $|\Omega(h,m)|<9$ .

This is a conclusion about every small return, not just one selected return. The next step needs precisely this quantifier: antisymmetrizing in two returns makes an integer pairing have absolute value strictly less than $1$ , so it vanishes. Successive minima then produce a rational fast space, and a nonzero return with a quantitative lower bound controls the effective curvature. Until these further steps are established, the present conclusion cannot be stated as (2).

This chapter uses two distinct source slots and two observation coordinates. A same-slot inner product requires a separate smoothing of the interval and a one-dimensional Fourier return argument. It does not follow simply by deleting one observation function.