Controlled ranges in the cubic collision count

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Status

The estimates below are proved. They remove several ranges at the desired $O(N^3)$ scale from the cubic energy route. They do not bound the full energy or establish positive density.

Fix positive integers $a_1,a_2,a_3$, and count pairs $\mathbf x,\mathbf y\in[1,N]^3\cap\mathbb Z^3$ with

$$F(\mathbf x)=F(\mathbf y),\qquad F(\mathbf x)=\sum_i a_i x_i^3.$$

Write $\mathbf h=\mathbf x-\mathbf y$. For $\mathbf h\ne0$, write $\mathbf h=t\mathbf k$, where $t>0$ and $\mathbf k$ is primitive. For every fixed $0<\eta<1/2$, the union of the following three ranges contributes $O_{\mathbf a,\eta}(N^3)$:

$$h_1h_2h_3=0, \qquad 0<\|\mathbf h\|_\infty\le N^{2/3-\eta}, \qquad \|\mathbf k\|_\infty\le N^{1/2-\eta}. \tag{1}$$

The last two ranges have power-saving bounds after coordinate-zero differences are removed.

For the special coefficients $(1,2,4)$, the later argument in Coordinate-zero differences also gives a power saving in the first range. It removes those inputs from the logarithmic-multiplicity problem at a cost of $o(N^3)$.

A uniform binary quadratic bound

Fix $L>0$ and $\varepsilon>0$. If $A,B,m$ are nonzero integers of absolute value at most $N^L$, with $N\ge2$, then

$$\#\{(u,v)\in[-N,N]^2\cap\mathbb Z^2:Au^2+Bv^2=m\} \ll_{L,\varepsilon}N^\varepsilon. \tag{2}$$

Set $D=-AB$ and $n=Am$. Multiplication by $A$ gives $(Au)^2-Dv^2=n$. If $D=d^2$ is a square, factorization into $(Au-dv)(Au+dv)=n$ bounds the count by $2\tau(|n|)$.

Otherwise put $K=\mathbb Q(\sqrt D)$. Each solution determines a distinct algebraic integer

$$\alpha=Au+v\sqrt D\in\mathcal O_K, \qquad N_{K/\mathbb Q}(\alpha)=n.$$

There are at most $\tau(|n|)$ integral ideals of norm $|n|$, uniformly in $K$. At a split prime the number of ideals of norm $p^e$ is $e+1$; at a ramified prime it is one; at an inert prime it is at most one. Multiply these bounds over primes.

The relevant generators of any one principal ideal are uniformly controlled. Both embeddings of $\alpha$ have absolute value at most

$$U=N(|A|+\sqrt{|AB|})\le2N^{L+1}.$$

In an imaginary quadratic field each ideal has at most six generators of a prescribed norm. In a real quadratic field generators differ by units, while their first absolute embedding lies between $|n|/U$ and $U$. A fundamental unit whose first absolute embedding exceeds one has that embedding at least $\varphi=(1+\sqrt5)/2$. Indeed, after adjusting its sign, a norm-one unit $z>1$ satisfies $z+z^{-1}\in\mathbb Z_{\ge3}$, and a norm-minus-one unit satisfies $z-z^{-1}\in\mathbb Z_{\ge1}$. Thus only $O_L(\log N)$ units can occur in the indicated interval, independently of the field.

We obtain $O_L(\tau(|Am|)\log N)$ solutions. The divisor bound, with its exponent chosen in terms of $L,\varepsilon$, proves (2).

A uniform ternary quadratic bound

If $ABC\ne0$ and $|A|,|B|,|C|,|M|\le N^L$, then

$$\#\{(u,v,w)\in[-N,N]^3\cap\mathbb Z^3: Au^2+Bv^2+Cw^2=M\} \ll_{L,\varepsilon}N^{1+\varepsilon}. \tag{3}$$

For fixed $w$, apply (2) to $m=M-Cw^2$ when it is nonzero, increasing the fixed height exponent as necessary. The exceptional equation $M-Cw^2=0$ has at most two integral solutions $w$. For each, the homogeneous binary equation has $O(N)$ solutions, since fixing $v$ permits at most two values of $u$. This proves (3), without a primitivity or nonsquare-discriminant hypothesis. Fixed constant multiples of $N$ in the coordinate bounds cause no change.

Applying the bound to cubic differences

Put $u_i=x_i+y_i$. The identity $4(x_i^3-y_i^3)=h_i(3u_i^2+h_i^2)$ gives

$$3\sum_i a_i h_i u_i^2=-\sum_i a_i h_i^3. \tag{4}$$

When $h_1h_2h_3\ne0$, equation (3) applies uniformly for each $\mathbf h$; its coefficients and target have polynomial height in $N$. Ignoring parity and positivity restrictions only enlarges the count. Consequently, for $1\le H\le N$,

$$E_{\mathrm{short}}(N,H) \ll_{\mathbf a,\varepsilon}N^{1+\varepsilon}H^3. \tag{5}$$

Taking $H=N^{2/3-\eta}$ and $\varepsilon<3\eta$ gives a power saving over $N^3$.

Alternatively, after $\mathbf h=t\mathbf k$ and division by $t$, equation (4) becomes

$$3\sum_i a_i k_i u_i^2=-t^2C(\mathbf k),\qquad C(\mathbf k)=\sum_i a_i k_i^3. \tag{6}$$

For $K=\|\mathbf k\|_\infty$ one has $t\le N/K$, and the target is $O_{\mathbf a}(N^2K)$, so (3) is uniform here too. Since $O(m^2)$ primitive vectors have maximum norm $m$,

$$E_{\mathrm{direction}}(N,K_0) \ll_{\mathbf a,\varepsilon} N^{1+\varepsilon}\sum_{\substack{\mathbf k\text{ primitive}\\ 0<\|\mathbf k\|_\infty\le K_0}} \frac{N}{\|\mathbf k\|_\infty} \ll_{\mathbf a,\varepsilon}N^{2+\varepsilon}K_0^2. \tag{7}$$

Coordinate-zero directions are excluded here and handled next. Taking $K_0=N^{1/2-\eta}$ and $\varepsilon<2\eta$ gives a power saving.

Coordinate-zero differences

Let $f(\alpha)=\sum_{n\le N}e(\alpha n^3)$. The number of collisions with $x_3=y_3$ is

$$N\int_0^1|f(a_1\alpha)f(a_2\alpha)|^2\,d\alpha \le N\int_0^1|f(\alpha)|^4\,d\alpha \ll N^3. \tag{8}$$

The inequality uses Cauchy--Schwarz and invariance of the integral under multiplication of $\alpha$ by a positive integer. The log-free fourth moment follows from Heath-Brown, The density of rational points on cubic surfaces, Theorem 1. Its statement was checked in the indexed primary text: for a nonsingular cubic surface with three rational coplanar lines, integer points off the rational lines have count $O_\varepsilon(N^{4/3+\varepsilon})$. The Fermat cubic surface meets these hypotheses; its finitely many lines contribute $O(N^2)$. Applying (8) to each coordinate proves the first bound in (1).

The remaining range and unsuccessful analytic transfers

For $(a_1,a_2,a_3)=(1,2,4)$, $C(\mathbf k)$ never vanishes at a nonzero rational vector: the nonzero summands have distinct $2$-adic valuations modulo three. Thus all remaining fibers are nonsingular. The part of the energy route not controlled by (1) is confined to

$$k_1k_2k_3\ne0,\qquad N^{1/2-\eta}<K\le N,\qquad tK>N^{2/3-\eta}. \tag{9}$$

This still contains the critical shell $K\asymp N$, $t=1$. Its expected $O(1)$ count per direction has not been proved on average. The weaker bounded-fiber criterion in the energy note could bypass a full energy bound; it is not established either.

Direct use of Browning--Heath-Brown, Counting rational points on quadric surfaces, Theorem 1.1 does not close this range. Even when its hypothesis on the squarefull part of the determinant holds, its bound contains $(\|Q\|^4/|\det Q|)^{5/8}$. For generic directions of size $K$, our four-variable $Q_{\mathbf k}$ has height of order $K^3$ and determinant of order $K^6$, giving a factor $K^{15/4}$. The resulting estimate is too large to sum. Replacing $t$ by $t'=Kt$ balances the real coefficients but introduces the sublattice $K\mid t'$ and rational coefficients; no estimate preserving the necessary arithmetic saving under this operation was established. This rejects the direct substitution into that theorem, not a sharper anisotropic average.

Fixing one coordinate in (6) gives another attempted transfer, through binary quadratic norm equations. After choosing two same-sign coefficients, the primitive diagonal positive binary form $[A,0,B]$ represents an ambiguous ideal class: its inverse $[A,-0,B]$ is the same class. Varying diagonal factorizations therefore does not average over the whole class group, and dividing an unrestricted ideal count by the class number is unjustified. The target values are also linked to $\mathbf k$, rather than running through a complete norm interval.

The unconditional result inspected in Browning--Munshi--Wang, Beyond the square-root barrier: cubic forms of Perazzo type, Theorem 1.1 concerns the singular fourfold $\sum_i x_i y_i^2=0$. The diagonal collision fourfold is smooth, so it is not linearly equivalent to that example. In difference coordinates the additional term $C(\mathbf h)$ has the same cubic scale as $3\sum_i a_i h_i u_i^2$ in the outstanding range; simply omitting it is not a valid application of the theorem. No perturbation or averaged estimate bridging this difference was proved.

The precise complete-sum change in the Perazzo transfer

A subsequent check identified the arithmetic step, rather than merely the geometric distinction, that prevents the direct transfer. The calculation below is self-contained. It supplies no new global energy estimate.

Let $p\nmid6a_1a_2a_3$, let $\chi$ be the quadratic character modulo $p$, extended by zero, and let $\tau_p=\sum_v\chi(v)e_p(v)$. Put

$$H(\mathbf h,\mathbf u)=C(\mathbf h)+3\sum_i a_i h_i u_i^2, \qquad C(\mathbf h)=\sum_i a_i h_i^3.$$

For frequencies $\boldsymbol\ell,\mathbf s\in\mathbb F_p^3$ with $s_1s_2s_3\ne0$, define

$$T_p=\sum_{\lambda\ne0}\sum_{\mathbf h,\mathbf u} e_p\bigl(\lambda H+\boldsymbol\ell\cdot\mathbf h+ \mathbf s\cdot\mathbf u\bigr).$$

Terms with some $h_i=0$ vanish on summing over $u_i$. Completing the three quadratic squares in the other terms gives exactly

$$T_p=\tau_p^3\chi(27a_1a_2a_3) \sum_{h_1h_2h_3\ne0}\chi(h_1h_2h_3)e_p(\boldsymbol\ell\cdot\mathbf h) \sum_{\lambda\ne0}\chi(\lambda) e_p\bigl(\lambda C(\mathbf h)-D(\mathbf h)/\lambda\bigr), \tag{10}$$

where

$$D(\mathbf h)=\sum_i\frac{s_i^2}{12a_i h_i},\qquad L(\mathbf h)=h_1h_2h_3D(\mathbf h).$$

If the cubic term is identically removed, the inner sum is $\tau_p\chi(-D)$, so (10) reduces to

$$T_p^{(0)}=p^2\chi(27a_1a_2a_3) \sum_{h_1h_2h_3\ne0}\chi(-L(\mathbf h)) e_p(\boldsymbol\ell\cdot\mathbf h). \tag{11}$$

Here $L$ is a nondegenerate ternary quadratic form: its three nonzero mixed-term coefficients are $s_i^2/(12a_i)$. For any nondegenerate ternary quadratic form $Q(\mathbf h)=\mathbf h^TM\mathbf h$, expansion of $\chi$ by its Gauss sum, followed by completing squares, gives

$$\sum_{\mathbf h}\chi(Q(\mathbf h))e_p(\boldsymbol\ell\cdot\mathbf h) =p\chi(-\det M) \begin{cases} p-1,&\boldsymbol\ell^TM^{-1}\boldsymbol\ell=0,\\ -1,&\boldsymbol\ell^TM^{-1}\boldsymbol\ell\ne0. \end{cases} \tag{12}$$

On each coordinate plane, $L$ is a nonzero constant times the product of the other two coordinates; its character Fourier sum has magnitude at most $p$. Intersections of two coordinate planes contribute zero. Thus (11) has magnitude at most $4p^3$ outside the dual quadratic condition in (12). This is the special finite-field cancellation behind the unperturbed calculation.

For $C(\mathbf h)\ne0$, the inner sum of (10) instead has the Salié identity

$$\sum_{\lambda\ne0}\chi(\lambda)e_p(C\lambda-D/\lambda) =\tau_p\chi(C)\sum_{v^2=-4CD}e_p(v). \tag{13}$$

For completeness, sum both sides against $e_p(tD)$ over $D$. At $t\ne0$, each sum is $p\chi(t)e_p(C/t)$; at $t=0$, each is zero. On the right this follows by one quadratic Gauss sum. This proves (13), including $D=0$.

Accordingly the part of $T_p$ with $C(\mathbf h)\ne0$ is

$$p^2\chi(27a_1a_2a_3) \sum_{\substack{h_1h_2h_3\ne0\\C(\mathbf h)\ne0}} \chi(h_1h_2h_3C(\mathbf h))e_p(\boldsymbol\ell\cdot\mathbf h) \sum_{v^2=-4C(\mathbf h)D(\mathbf h)}e_p(v). \tag{14}$$

Directions with $C=0$ give the corresponding restricted part of (11). Recovering a $p^3$ estimate now requires an $O(p)$ bound for the combined sum after its $p^2$ prefactor is removed. Formula (12) does not supply that bound: the cubic term remains in the phase and in the square-root condition of (14). No needed cancellation estimate or subsequent modulus average was proved. This rules out the direct substitution into the unperturbed proof, not every possible treatment of the perturbed sum.

Coordinate-zero differences have a power saving for distinct cube classes

The following strengthens the first range of (1). The external input is Salberger, Counting rational points on projective varieties (2023), Corollary 0.7, pp. 1094–1095. The theorem and its projective-height convention were checked in the full paper. For a diagonal degree-$d$ surface in $\mathbb P^3$ with nonzero rational coefficients, its rational points of height at most $B$ outside all two-term-zero-subsum loci number $O_d(B^{3/\sqrt d}(\log B)^4+1)$, uniformly in the coefficients.

Let $a,b>0$ be rational, with $a/b\notin\mathbb Q^{\times3}$. Then

$$\#\{(x,y,u,v)\in[1,N]^4\cap\mathbb Z^4: ax^3+by^3=au^3+bv^3,\ (x,y)\ne(u,v)\} \ll N^{\sqrt3}(1+\log N)^4. \tag{15}$$

Indeed, a same-sign two-term subsum cannot vanish. A subsum $ax^3-au^3=0$ forces $x=u$ and then $y=v$, and likewise for the other matching pair. A crossed subsum, such as $ax^3-bv^3=0$, would make $a/b$ a rational cube. Thus every primitive nontrivial positive solution lies in Salberger's open subset. Splitting a general solution by its coordinate gcd gives

$$\sum_{g\le N}O\bigl((N/g)^{\sqrt3}(1+\log(N/g))^4+1\bigr) \ll N^{\sqrt3}(1+\log N)^4,$$

since $\sqrt3>1$. This proves (15).

Consequently, if the three coefficients of $F$ have distinct rational cube classes, the number $E_0(N)$ of off-diagonal collisions having at least one equal coordinate satisfies

$$E_0(N)\ll N^{1+\sqrt3}(1+\log N)^4=o(N^3). \tag{16}$$

For each choice of the common coordinate, sum (15) over its $N$ possible values. In particular, (16) applies to $F=x^3+2y^3+4z^3$.

Removal from the logarithmic moment

For this special form, let $Z_N$ be the inputs in $[1,N]^3$ having a distinct equal-output partner with some equal coordinate. Let $g_N(n)$ count the representations remaining after deleting $Z_N$. Then $|Z_N|\le E_0(N)$, and each remaining fiber is coordinatewise injective; in particular $g_N(n)\le N$.

Writing $R_N(n)=g_N(n)+b_N(n)$, use $R_N(n)\le N^2$ and $\log(1+t)\le t$ to obtain, also when $g_N(n)=0$,

$$0\le R_N(n)\log R_N(n)-g_N(n)\log g_N(n) \le b_N(n)(2\log N+1).$$

Here $0\log0=0$. Since $\sum_n b_N(n)=|Z_N|$, (16) implies

$$\sum_n\bigl(R_N(n)\log R_N(n)-g_N(n)\log g_N(n)\bigr) \ll N^{1+\sqrt3}(1+\log N)^5=o(N^3). \tag{17}$$

Thus deleting these inputs preserves the bounded-logarithmic-moment question, including the effect on their surviving fiber-mates. The remaining fibers, whose representations differ in all three coordinates, are not bounded by this argument. No density conclusion follows from (15)–(17) alone.

Near neighbors for arbitrary positive coefficients

There is also a useful conclusion without the distinct-cube-class hypothesis. Let $B_N^{(0)}(\varepsilon)$ count inputs having a distinct equal-output partner within Euclidean distance $\varepsilon N$, with at least one equal coordinate. Then for every positive ternary diagonal cubic,

$$B_N^{(0)}(\varepsilon) \le6\varepsilon N^3+3N^2 +O\bigl(N^{1+\sqrt3}(1+\log N)^4\bigr). \tag{18}$$

The bound remains valid in a fixed interior box.

To prove this, fix the common coordinate $i$ and write $j,k$ for the other two. Salberger's open subset contributes the power-saving term as above, even if the coefficient ratio is a cube. The only remaining off-diagonal positive solutions have

$$y_j=cx_k,\qquad y_k=x_j/c,\qquad c=(a_k/a_j)^{1/3}\in\mathbb Q.$$

Indeed, the same-sign two-term subsums cannot vanish, the same-coordinate subsum forces the diagonal, and either crossed subsum forces this weighted swap. The distance condition implies $|x_j-cx_k|\le\varepsilon N$. Fixing $x_i,x_k$ leaves at most $2\varepsilon N+1$ possibilities for $x_j$. Sum over the three choices of $i$ to prove (18).

It follows that

$$\lim_{\varepsilon\downarrow0}\limsup_{N\to\infty} B_N^{(0)}(\varepsilon)/N^3=0.$$

Thus the near-neighbor criterion may now restrict entirely to partners differing in all three coordinates, for every positive coefficient tuple. That remaining existence estimate, at distance $\varepsilon N$ with fixed $\varepsilon$, is still unproved.

A uniform bound on each secant-ratio slice

The following additional estimates are proved. They control a new arithmetic stratum of the near-neighbor count, but not all its inputs.

Write an ordered nontrivial collision as

$$F(x)=F(x+tk),\qquad t\ge1,\quad k\in\mathbb Z^3\text{ primitive},$$

and put

$$K=\|k\|_\infty,\quad C=F(k),\quad B=\sum_i a_i x_i k_i^2,\quad A=\sum_i a_i x_i^2k_i.$$

Throughout this section assume $k_1k_2k_3C\ne0$. For the special form $x^3+2y^3+4z^3$, the condition $C\ne0$ is automatic.

Slice lemma. For every $q\in\mathbb Q$, the number of $x\in[1,N]^3\cap\mathbb Z^3$ satisfying

$$B=qC,\qquad F(x+tk)=F(x)$$

is

$$O_{\mathbf a}(1+N/K), \tag{19}$$

uniformly in $q,k,t,N$. Requiring $x+tk\in[1,N]^3$ can only decrease this count.

Proof. The collision equation is $3A+3tB+t^2C=0$. Set $z=x-qk$. The two equations become

$$\sum_i a_i k_i^2z_i=0,\qquad \sum_i a_i k_i z_i^2 =-\left(q^2+tq+\frac{t^2}{3}\right)C. \tag{20}$$

The scalar in parentheses is $(q+t/2)^2+t^2/12>0$.

Let $n=(a_i k_i^2)_i$, $M=\operatorname{diag}(a_i k_i)$, and $W=n^\perp$. The quadratic form $z^TMz$ is nondegenerate on $W$. Indeed, a vector $z$ in the radical of its restriction has $Mz\in\mathbb Rn$, so $z\in\mathbb Rk$. Membership in $W$ and $n\cdot k=C\ne0$ then force $z=0$. Its nonzero level in (20) is therefore a nonsingular affine conic and contains no line. Equivalently, the homogeneous ternary matrix of that conic has rank three.

If the affine plane $B=qC$ has an integral point, its integral points are a coset of the rank-two lattice

$$\Lambda=\{v\in\mathbb Z^3:n\cdot v=0\},\qquad \det\Lambda=\frac{\|n\|_2}{\gcd(a_1k_1^2,a_2k_2^2,a_3k_3^2)} \asymp_{\mathbf a}K^2. \tag{21}$$

For each prime, some $k_i$ is a unit; hence the denominator in (21) divides $\operatorname{lcm}(a_1,a_2,a_3)$.

Choose a shortest nonzero lattice vector $v$. It is primitive in $\Lambda$, and the two-dimensional lattice bound gives $\|v\|\ll\sqrt{\det\Lambda}$. Extend it to a lattice basis. The parallel lattice lines in direction $v$ are spaced $\det\Lambda/\|v\|\gg\sqrt{\det\Lambda}\asymp_{\mathbf a}K$. Only $O_{\mathbf a}(1+N/K)$ such lines meet the box. Each meets the conic at most twice, proving (19). $\square$

Fixed normalized ratios and bounded denominators

Define the normalized secant ratio

$$\rho=\frac{B}{tC}.$$

It determines the third intersection of the secant with the cubic: its parameter on the line $x+s(tk)$ is $s=-1-3\rho$.

For a prescribed rational $\rho$, let $E_\rho(N,H)$ count ordered collisions in $[1,N]^3$ with $0<tK\le H$ and $k_1k_2k_3C\ne0$. For $1\le H\le N$,

$$E_\rho(N,H)\ll_{\mathbf a}H^3+NH^2 \ll_{\mathbf a}NH^2, \tag{22}$$

with a constant independent of $\rho$. For each $k,t$, apply (19) with $q=t\rho$. There are $O(m^2)$ primitive directions of height $m$, so the resulting sum is bounded by

$$\sum_{m\le H}m^2\frac Hm\left(1+\frac Nm\right) \ll H^3+NH^2.$$

There is also a bound allowing infinitely many normalized ratios as $N$ varies. Fix $D\ge1$ and $0<\eta\le A_0:=\sum_i a_i$. Let $E_{D,\eta}(N,H)$ count the same pairs, restricted by

$$|F(k)|\ge\eta K^3, \qquad \operatorname{den}\!\left(\frac{B}{tC}\right)\le D,$$

where $\operatorname{den}$ means the reduced positive denominator. Then

$$E_{D,\eta}(N,H)\ll_{\mathbf a} \frac{D^2}{\eta}N^2H. \tag{23}$$

To prove this, partition the displacements into dyadic shells $J/2<tK\le J$. Since $B\le A_0NK^2$, a contributing ratio satisfies

$$|\rho|\le\frac{2A_0N}{\eta J}=:L.$$

There are at most $\sum_{d\le D}(2Ld+1)\ll_{\mathbf a}D^2N/(\eta J)$ such rationals of denominator at most $D$. Apply (22), with $H=J$, to each. This bounds the shell by $O_{\mathbf a}(D^2N^2J/\eta)$. The upper endpoints of the shells up to $H$ sum to at most $2H$, proving (23).

For fixed $D,\eta$, the full collision energy on this stratum is $O_{\mathbf a,D,\eta}(N^3)$. Its near-neighbor existence count at Euclidean distance at most $\varepsilon N$ is at most

$$O_{\mathbf a}\!\left(D^2\varepsilon N^3/\eta\right), \tag{24}$$

because every such input participates in an ordered pair counted by (23). Thus this stratum satisfies the vanishing near-neighbor limit required in the density criterion.

Remaining gap. No uniform tail estimate permits $D\to\infty$ or $\eta\downarrow0$ while retaining almost all relevant input mass. Ratios can have reduced denominators growing with $N$, and directions can have $|F(k)|/K^3$ arbitrarily small. The denominator in (23) is that of $B/(tC)$, not the denominator $C/\gcd(C,3B)$ of the third secant point in primitive projective coordinates. The bounds do not imply the full near-neighbor estimate, bounded typical multiplicity, or positive density.

Binary norm ratios of small height

This estimate concerns $F=x^3+y^3+cz^3$ with fixed $c\in\mathbb N$, not the distinct-coefficient form $x^3+2y^3+4z^3$. Put $Q(x,y)=x^2-xy+y^2$. Let $E_D(N)$ count ordered collisions

$$F(x,y,z)=F(u,v,w),\qquad x,y,z,u,v,w\in[1,N]\cap\mathbb Z,$$

such that $Q(u,v)/Q(x,y)=p/q$ in lowest terms with $\max(p,q)\le D$, excluding the identical triple and the swap $(u,v,w)=(y,x,z)$. For all $N\ge2$, $D\ge1$, and $\varepsilon>0$,

$$E_D(N)\ll_{c,\varepsilon}D N^{2+\varepsilon}. \tag{25}$$

Write $m=Q(x,y)$, $m'=Q(u,v)$ and $H=x^3+y^3-u^3-v^3$. First suppose $H\ne0$. Fix $(x,y)$. Since $q\mid m$, at most $D\tau(m)$ pairs $(p,q)$ can occur, and each determines $m'=pm/q$. All contributing norms are at most $N^2$. The elementary norm bound

$$\#\{(u,v)\in\mathbb Z^2:Q(u,v)=m'\}\le6\tau(m')$$

follows in the Eisenstein integers: there are at most $\tau(m')$ integral ideals of norm $m'$, and each principal ideal has six generators. For each resulting binary quadruple, the equation

$$(w-z)(w^2+wz+z^2)=H/c\ne0$$

has no solutions unless $c\mid H$, and otherwise at most $2\tau(|H/c|)$ solutions. Fixing the nonzero difference $w-z$ leaves a quadratic equation. Since $|H|\le2N^3$, the divisor bound, with its exponents chosen sufficiently small, proves (25) for $H\ne0$.

If $H=0$, then $z=w$. Write $s=x+y$, $t=u+v$, $d=x-y$, $e=u-v$. From $sm=tm'$ and $m'/m=p/q$, we obtain

$$s=p\ell,\qquad t=q\ell,\qquad 1\le\ell\le\frac{2N}{\max(p,q)}.$$

The identities $4m=s^2+3d^2$ and $4m'=t^2+3e^2$ give

$$3q e^2-3p d^2=(p^3-q^3)\ell^2. \tag{26}$$

If $p=q$, then $p=q=1$ and equal trace and norm force only the excluded identical pair and swap. Otherwise the right side of (26) is nonzero. The coefficients and target have polynomial height in $N$, since $p,q\le2N$ here. The uniform binary quadratic bound gives $O_\varepsilon(N^\varepsilon)$ choices of $(d,e)$ per $(p,q,\ell)$. Thus the binary quadruples number at most

$$\ll_\varepsilon N^\varepsilon \sum_{k\le\min(D,2N)}2k\left\lfloor\frac{2N}{k}\right\rfloor \ll_\varepsilon D N^{1+\varepsilon}.$$

The $N$ choices $z=w$ complete the proof of (25).

For any fixed $0<\eta<1$, take $D=N^{1-\eta}$ and $\varepsilon<\eta$. Removing all endpoints of these nontrivial collisions costs $o(N^3)$ inputs, and also $o(N^3)$ in the logarithmic moment, since $R_N(n)\le N^2$. The unrestricted norm ratios can have height of order $N^2$, however. No bound for their complementary input mass has been proved. In particular (25) does not bound the cross-coordinate correlation or conditional entropy.

Nearly vanishing directions near a balanced positive ray

The following geometric observation is checked. It does not extend the exact coordinate-sharing count to nearby directions.

For $F=\sum_{i=1}^3 a_i x_i^3$, $a_i>0$, put $X_i=a_i^{1/3}x_i$, $H_i=a_i^{1/3}h_i$ and $R=\max_i|H_i|>0$. Assume

$$|X_i-M|\le\delta M,\qquad R\le\varepsilon M,\qquad F(\mathbf x+\mathbf h)=F(\mathbf x),$$

where $\delta+\varepsilon$ is sufficiently small. Expanding the collision identity gives

$$S:=H_1+H_2+H_3=O((\delta+\varepsilon)R).$$

Indeed the linear term is $3M^2S+O(\delta M^2R)$, while the quadratic and cubic terms are $O(MR^2+R^3)$. Using

$$\sum_iH_i^3=3H_1H_2H_3+ S\left(\sum_iH_i^2-\sum_{i<j}H_iH_j\right),$$

we conclude that

$$|F(\mathbf h)|\le\eta R^3 \quad\Longrightarrow\quad \min_i|H_i|\ll(\eta+\delta+\varepsilon)R. \tag{27}$$

For the last implication, $|S|\le R/2$ makes the largest two $|H_i|$ comparable, and the product is $O((\eta+\delta+\varepsilon)R^3)$.

Thus nearly vanishing directions in this cone approach weighted coordinate-swap directions. But for nonzero integer differences with $\|h\|_\infty\le H$ and one coordinate at most $L$, the available ternary quadratic bound gives only $O_{\mathbf a,\nu}(N^{1+\nu}H^2L)$. At $H$ proportional to $N$ and $L$ a fixed small fraction of $H$, this is not $o(N^3)$. Continuity of the real geometry therefore does not justify replacing the uncontrolled thin sector by the already controlled exact coordinate-zero locus. No near-neighbor existence estimate follows.

Binary trace ratios, uniformly in the prescribed ratio

Fix $c\in\mathbb N$ and put $F(x,y,z)=x^3+y^3+cz^3$. Exclude the collisions $(u,v,w)=(x,y,z)$ and $(y,x,z)$. For every reduced positive rational $p/q$, the remaining collisions in $[1,N]^6$ with

$$\frac{u+v}{x+y}=\frac pq$$

number

$$O_{c,\varepsilon}(N^{2+\varepsilon}), \tag{28}$$

uniformly in $p,q$, even when their height grows with $N$.

Only $p,q\le2N$ can contribute. Write

$$x+y=q\ell,\quad u+v=p\ell,\quad d=x-y,\quad e=u-v.$$

Then

$$H:=4c(w^3-z^3) =\ell\bigl((q^3-p^3)\ell^2+3qd^2-3pe^2\bigr). \tag{29}$$

Suppose first that $z\ne w$. Fixing $(z,w)$ fixes a nonzero $H=O_c(N^3)$, and $\ell\mid H$. Unless

$$H/\ell-(q^3-p^3)\ell^2=0, \tag{30}$$

the uniform binary quadratic bound (2) gives $O_{c,\varepsilon}(N^\varepsilon)$ choices of $(d,e)$ for each divisor $\ell$. All coefficients and targets have polynomial height in $N$. The divisor bound, with smaller intermediate exponents, proves (28) for this part.

Under (30), one has $qd^2=pe^2$. If $p/q$ is not a rational square, then $d=e=0$. Enumerate $\ell\le2N/\max(p,q)$ instead. Equation (29) prescribes a nonzero difference of cubes, which has $O_{c,\varepsilon}(N^\varepsilon)$ solutions $(z,w)$: factor $w^3-z^3=(w-z)(w^2+wz+z^2)$ and fix its first factor.

If $p=a^2,q=b^2$ with $(a,b)=1$, set $A=\max(a,b)$. The homogeneous solutions are

$$d=at,\qquad e=\pm bt,\qquad \ell\le2N/A^2,\quad |t|\le N/A.$$

Here $A\le\sqrt{2N}$ whenever $\ell$ exists. Thus there are $O(N^2/A^3)$ pairs $(\ell,t)$. Each again prescribes a nonzero difference of cubes, giving $O_{c,\varepsilon}(N^{2+\varepsilon})$ collisions. The case $p=q=1$ cannot satisfy (30) when $H\ne0$.

Finally let $z=w$. If $p=q$, then $p=q=1$, and equal binary trace and cubic sum give only the excluded identical pair and swap. Otherwise, for each $\ell\le2N/\max(p,q)$, (29) is a binary quadratic equation with nonzero target. Apply (2) and then allow the $N$ choices of $z=w$. This completes the proof.

There are $O(D^2)$ positive rational ratios of height at most $D$, so their nontrivial collisions number

$$O_{c,\varepsilon}(D^2N^{2+\varepsilon}). \tag{31}$$

For every fixed $0<\eta<1/2$, taking $D=N^{1/2-\eta}$ and $\varepsilon<2\eta$ permits removal of all endpoints of these collisions at cost $o(N^3)$. The same is true for their contribution to the logarithmic moment, by the deletion inequality (17) and $R_N(n)\le N^2$.

Unlike the small-direction range, (28) permits unrestricted displacements. Unlike (25), it imposes no small-height condition on the binary norm ratio. Neither observation controls its complementary trace-ratio tail. A fiber can have many distinct traces of size comparable to $N$ with pairwise small gcd, so ordinary pigeonhole arguments do not supply a ratio of height at most $N^{1/2-\eta}$. No typical-multiplicity estimate or density conclusion follows.