Rudnick–Sarnak–Zaharescu Diophantine criterion for Poissonian statistics

For $\alpha\in[0,1]$, say that $\alpha$ is of type $\omega$ if there are only finitely many pairs $(a,q)$ satisfying $\lvert\alpha-a/q\rvert<q^{-\omega}$. Let $a/q$ range over the convergents to $\alpha$, let $\widetilde q$ denote the square-free part of $q$, and let $W_{\alpha,L,N}$ be the random variable counting the points $\alpha n^2\operatorname{mod}1$ in a random interval of length $L/N$. Rudnick–Sarnak–Zaharescu conjecture. If $\alpha$ is of type $2+\varepsilon$ for all $\varepsilon>0$ and its convergents satisfy

$$\lim_{q\to\infty}\frac{\log\widetilde q}{\log q}=1,$$

then, for every fixed $L>0$,

$$W_{\alpha,L,N}\xrightarrow{\operatorname{dist}}\operatorname{Po}(L)$$

as $N\to\infty$. This gives a Diophantine condition intended to ensure Poissonian local statistics for the quadratic sequence; the source presents it as a further conjecture and supplies no resolution evidence.