Local Law for Singular Values of Oscillatory Matrices

Abstract

Abstract We continue the study of spectra of oscillatory random matrices with fully dependent rows. Motivated by the $d$-dimensional skew-shift dynamics from ergodic theory, we introduce the $N\times N$ random matrices $$ \begin{align*} &X_{j,k}=\exp\left(2\pi \textrm{i} \sum_{q=1}^d\ \omega_{j,q} k^q\right), \end{align*}$$where $\{\omega _{j,q}\}_{1\leq j\leq N, 1\leq q\leq d}$ is a collection of i.i.d. random variables and $d$ is a fixed integer. We prove that as $N\to \infty $ the distribution of singular values converges to the local Marchenko–Pastur law up to scales $N^{-\theta _d}$ for an explicit $\theta _d>0$, as long as $d\geq 18$. Our approach provides a novel mechanism—deterministic oscillatory cancellations—for universal spectral laws. The proof blends techniques from random matrix theory, harmonic analysis, and analytic number theory such as strong estimates on the number of solutions to Diophantine equations in the form of Vinogradov’s main conjecture, proved by Bourgain–Demeter–Guth.