Local moment conjecture for exponential sums

For $d\geq 2$, let $S_{a,d}(x,N)=\sum_{n=1}^N a_n e(x_1n+\cdots+x_dn^d)$ and let $\rho_d$ be

$$\rho_d=\begin{cases}\frac{3d^2-4}{4},&d\text{ even},\\frac{3d^2-3}{4},&d\text{ odd}.\end{cases}$$

Local moment conjecture. For every $j\geq 0$,

$$2^{j\frac{d+1}{2}}\int_{[0,2^{-j}]^d}|S_{a,d}(x,N)|^p\,dx\lesssim_\epsilon\|a\|_{\ell^2}^p\begin{cases}N^\epsilon,&0<p\leq\rho_d,\\N^{\frac{p-\rho_d}{2}+\epsilon},&p>\rho_d\end{cases}.$$

This local estimate is intended as a strategy for the Fourier-decay conjecture and is sharp under the constructive-interference examples discussed in the paper; it remains open in general.