Rudnick–Sarnak–Heath-Brown conjecture on pair correlation of quadratic fractional parts

Let $\beta otin\frac{\text{?}}{}$ be an irrational real number. The pair correlation of the fractional parts of $n^2\beta$ is called Poissonian when it has the limiting statistics of a Poisson process. A real number $\beta$ is Diophantine if it satisfies the Diophantine approximation condition intended in the source.

Rudnick–Sarnak–Heath-Brown conjecture. Assume $\beta$ is Diophantine. Then the pair correlation for the fractional parts of $n^2\beta$ is Poissonian.

The conjecture seeks an explicit class of $\beta$ for which Poissonian pair correlation holds; the source notes that the result is known for almost all $\beta$, but no specific $\beta$ was known in the stated context.