# A Local Law for Singular Values from Diophantine Equations

@article{Adhikari2020ALL, title={A Local Law for Singular Values from Diophantine Equations}, author={Arka Adhikari and Marius Lemm}, journal={arXiv: Probability}, year={2020} }

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…

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