Decoupling conjecture for equally distributed points on a monomial curve

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Let d≥3d\geq 3 be an integer, let Γ={(ξ,ξd)∈R2:1≤∣ξ∣≤2}\Gamma=\{(\xi,\xi^d)\in\mathbb R^2:1\leq|\xi|\leq2\}, and let Pj=(ξj,ξjd)P_j=(\xi_j,\xi_j^d) for j=1,…,Nj=1,\ldots,N be equally distributed in Γ\Gamma. For each P∈ΓP\in\Gamma, let ωP\omega_P be an N−1×N−dN^{-1}\times N^{-d} rectangle centered at PP whose long side is parallel to the tangent line to Γ\Gamma at PP, let ΩN={ωP1,…,ωPN}\Omega_N=\{\omega_{P_1},\ldots,\omega_{P_N}\}, and suppose fω^=f^ 1ω\widehat{f_\omega}=\widehat f\,\mathbf 1_\omega. For every ε>0\varepsilon>0 and p≥2(d+1)p\geq2(d+1), the decoupling conjecture asserts that

∥∑ω∈ΩNfω∥p≲N12−d+1p+ε(∑ω∈ΩN∥fω∥p2)12.\left\|\sum_{\omega\in\Omega_N}f_\omega\right\|_p\lesssim N^{\frac12-\frac{d+1}{p}+\varepsilon}\left(\sum_{\omega\in\Omega_N}\|f_\omega\|_p^2\right)^{\frac12}.

This is a decoupling formulation associated with the discrete restriction problem above. The source presents it as a standard consequence of that conjecture and does not state whether it has been resolved.

References

Primary source

Xiaochun Li, “A Stein-Tomas type estimate and a decoupling inequality”, arXiv:2309.12835 (2023).

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