2 problems

For each $n$, let $A_n\subseteq[-\pi,\pi)$ be a set, and call the sequence $(A_n)$ shrinking when its size tends to zero. Let $\mathcal{N}_{A_n}$ denote the CUE eigenvalue count in…

Let $U$ be an $N\times N$ random matrix from the circular unitary ensemble with eigenvalues $\{e^{i\theta_j}\}_{1\le j\le N}$, where $0\le \theta_1\le\dots\le\theta_N<2\pi$. Choose…