The L^12 square-root cancellation conjecture for nondegenerate curves in R^4

Let $\phi_3,\phi_4$ be real analytic functions on $(0,3)$ satisfying the derivative bounds and nondegeneracy conditions

\|\phi_k'\|_{C^3}=\sum_{1\le n\le 4}\max_{1/2\le t\le 1}|\phi_k^{(n)}(t)|\le A_1,\quad k\in\{3,4\},

A_2\le \left|\det\begin{bmatrix}\phi_3^{(3)}(t)&\phi_3^{(4)}(t)\\ \phi_4^{(3)}(s)&\phi_4^{(4)}(s)\end{bmatrix}\right|\le A_3,\quad t,s\in[1/2,1],

|\phi_3^{(3)}(t)|\ge A_4,\quad t\in[1/2,1].

For a finite interval $I\subset {\mathbb Z}$ , define

{\mathcal E}_{I,N}(x)=\sum_{n\in I}e(nx_1+n^2x_2+\phi_3(n/N)x_3+\phi_4(n/N)x_4).

Assume $\alpha\ge\beta\ge0$ and $\alpha+\beta=3$ , and set $\omega_3=[0,N^\alpha]$ and $\omega_4=[0,N^\beta]$ . The $L^{12}$ square-root cancellation conjecture. One should have

\int_{[0,1]\times[0,1]\times\omega_3\times\omega_4}|{\mathcal E}_{[N/2,N],N}(x)|^{12}\,dx\lesssim_\epsilon N^{9+\epsilon}.

This conjecture extends sharp $L^{12}$ estimates for exponential sums associated with nondegenerate curves in ${\mathbb R}^4$ and would establish the expected square-root cancellation uniformly over the indicated anisotropic frequency boxes. Its status is not resolved in the supplied source.

A 2021 preprint formulates the conjecture for real-analytic nondegenerate curves in $\mathbb{R}^4$ , with $\alpha\ge\beta\ge0$ and $\alpha+\beta=3$ . The exponent $9$ is asserted to be optimal.

Known results

Bourgain settled the endpoint $(\alpha,\beta)=(2,1)$ .
The endpoint $(\alpha,\beta)=(3,0)$ follows from the Main Conjecture in Vinogradov’s Mean Value Theorem.
The 2021 preprint proves the estimate for $\frac{3}{2}\le\alpha\le2$ , with $\beta=3-\alpha$ .
For power curves, rescaling gives further results, including the moment curve $(t^3,t^4)$ .

May 2026 moment-curve development

A 2026 preprint claims to verify Demeter’s conjecture for the moment curve $\Phi(t)=(t,t^2,t^3,t^4)$ , covering the remaining $p=12$ range. This does not establish the stated conjecture for all real-analytic nondegenerate curves.

Current status (as of August 2026): The conjecture is settled for $\frac{3}{2}\le\alpha\le2$ and at $\alpha=3$ , with a 2026 preprint covering the remaining range for the moment curve; the general-curve range $2<\alpha<3$ remains open.

The $L^{12}$ square-root cancellation conjecture for nondegenerate curves in ${\mathbb R}^4$

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Progress summary

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