4 problems

Let $\beta$ be the inverse temperature of a Gaussian $\beta$-ensemble with $n$ particles, and let the separation be the smallest distance between two distinct particles. The cited…

Minimal-spacing conjecture. As $n\to\infty$,

Let $\alpha$ be irrational and let $d\geq2$, and consider the sequence $\alpha n^d$ modulo one. A real number $\alpha$ is badly approximable by rationals if its rational approximat…

Linear pair-correlation conjecture. For almost every $\lambda\in(\frac{1}{2},1)$, there exist $c,C>0$ such that