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The L12L^{12} square-root cancellation conjecture for nondegenerate curves in R4{\mathbb R}^4

From papers

Let ϕ3,ϕ4\phi_3,\phi_4 be real analytic functions on (0,3)(0,3) satisfying the derivative bounds and nondegeneracy conditions

∥ϕk′∥C3=∑1≤n≤4max⁡1/2≤t≤1∣ϕk(n)(t)∣≤A1,k∈{3,4},\|\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\}, A2≤∣det⁡[ϕ3(3)(t)ϕ3(4)(t)ϕ4(3)(s)ϕ4(4)(s)]∣≤A3,t,s∈[1/2,1],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], ∣ϕ3(3)(t)∣≥A4,t∈[1/2,1].|\phi_3^{(3)}(t)|\ge A_4,\quad t\in[1/2,1].

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

EI,N(x)=∑n∈Ie(nx1+n2x2+ϕ3(n/N)x3+ϕ4(n/N)x4).{\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 α≥β≥0\alpha\ge\beta\ge0 and α+β=3\alpha+\beta=3, and set ω3=[0,Nα]\omega_3=[0,N^\alpha] and ω4=[0,Nβ]\omega_4=[0,N^\beta]. The L12L^{12} square-root cancellation conjecture. One should have

∫[0,1]×[0,1]×ω3×ω4∣E[N/2,N],N(x)∣12 dx≲ϵN9+ϵ.\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 L12L^{12} estimates for exponential sums associated with nondegenerate curves in R4{\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.

Progress summary

Partially solved

The conjecture is proved in substantial parameter ranges and for the moment curve, but the full statement for general curves remains open.

A 2021 preprint formulates the conjecture for real-analytic nondegenerate curves in R4\mathbb{R}^4, with α≥β≥0\alpha\ge\beta\ge0 and α+β=3\alpha+\beta=3. The exponent 99 is asserted to be optimal.

Known results

  • Bourgain settled the endpoint (α,β)=(2,1)(\alpha,\beta)=(2,1).
  • The endpoint (α,β)=(3,0)(\alpha,\beta)=(3,0) follows from the Main Conjecture in Vinogradov’s Mean Value Theorem.
  • The 2021 preprint proves the estimate for 32≤α≤2\frac{3}{2}\le\alpha\le2, with β=3−α\beta=3-\alpha.
  • For power curves, rescaling gives further results, including the moment curve (t3,t4)(t^3,t^4).

May 2026 moment-curve development

A 2026 preprint claims to verify Demeter’s conjecture for the moment curve Φ(t)=(t,t2,t3,t4)\Phi(t)=(t,t^2,t^3,t^4), covering the remaining p=12p=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 32≤α≤2\frac{3}{2}\le\alpha\le2 and at α=3\alpha=3, with a 2026 preprint covering the remaining range for the moment curve; the general-curve range 2<α<32<\alpha<3 remains open.

Sources
Sources & referencesView supporting material

Primary source

Ciprian Demeter, “On L^12 square root cancellation for exponential sums associated with nondegenerate curves in R^4”, arXiv:2101.08220 (2021).

decouplingexponential-sumsharmonic-analysis

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