Bourgain–Rudnick–Sarnak variance conjecture for lattice points on the sphere

Let $\mathcal N=\{n\in\mathbb N:n\not\equiv 0,4,7\pmod{8}\}$, let $\mathcal E(n)=\{\mathbf{x}\in\mathbb Z^3:|\mathbf{x}|^2=n\}$, let $\widehat{\mathcal E}(n)=n^{-1/2}\mathcal E(n)\subset\mathbb S^2$, and write $N_n=\#\mathcal E(n)$. For a spherical cap $\Omega\subset\mathbb S^2$, define $Z(n,\Omega)=\#(\widehat{\mathcal E}(n)\cap\Omega)$ and

$$\operatorname{Var}(\Omega,n)=\int_{\mathbb S^2}\left|Z(n,\Omega+\zeta)-N_n\sigma(\Omega)\right|^2\,d\sigma(\zeta),$$

where $\sigma$ is normalized area measure on $\mathbb S^2$. Bourgain–Rudnick–Sarnak variance conjecture. Let $\Omega_n$ be a sequence of spherical caps. If $N_n^{-1+\varepsilon}\ll\sigma(\Omega_n)\ll N_n^{-\varepsilon}$ as $n\to\infty$, with $n\in\mathcal N$, then

$$\operatorname{Var}(\Omega_n,n)\sim N_n\sigma(\Omega_n).$$

The conjecture predicts the natural-size variance for spherical-cap counts in the equidistributed sets of lattice points on expanding spheres. The paper states that Bourgain, Rudnick, and Sarnak posed this asymptotic and proves an upper bound of the correct size on average over the radii; the full pointwise asymptotic remains unresolved in the stated range.