  We introduce the $N\times N$ random matrices $$ X_{j,k}=\exp\left(2\pi i
\sum_{q=1}^d\ \omega_{j,q} k^q\right) \quad \text{with }
\{\omega_{j,q}\}_{\substack{1\leq j\leq N\\ 1\leq q\leq d}} \text{ i.i.d.
random variables}, $$ and $d$ a fixed integer. We prove that the distribution
of their singular values converges to the local Marchenko-Pastur law at scales
$N^{-\theta_d}$ for an explicit, small $\theta_d>0$, as long as $d\geq 18$. To
our knowledge, this is the first instance of a random matrix ensemble that is
explicitly defined in terms of only $O(N)$ random variables exhibiting a
universal local spectral law. Our main technical contribution is to derive
concentration bounds for the Stieltjes transform that simultaneously take into
account stochastic and oscillatory cancellations. Important ingredients in our
proof are strong estimates on the number of solutions to Diophantine equations
(in the form of Vinogradov's main conjecture recently proved by
Bourgain-Demeter-Guth) and a pigeonhole argument that combines the Ward
identity with an algebraic uniqueness condition for Diophantine equations
derived from the Newton-Girard identities.
