Fine-scale distribution of roots of quadratic congruences

The candidate solution restates the two main claims of the paper: (i) existence of a D-dependent limiting point process Ξ for normalized roots and convergence Ξ_{N,λ} ⇒ Ξ for absolutely continuous λ (Theorem 1.2), and (ii) convergence of the pair correlation measure with an even, continuous density w_D (Theorem 1.1). It follows the paper’s geometric encoding of roots as tops of closed geodesics with apex μ/m + i√D/m (Theorem 5.1) and invokes equidistribution on Γ\G to obtain the limit process; definitions of Ξ_{N,λ} and the endpoint-continuity of limiting finite-dimensional distributions match the paper verbatim. Minor phrasing differences (geodesic vs horocycle flow emphasis) do not affect correctness. See Theorem 1.2 and (1.5) for Ξ_{N,λ} and the limit statement, and Theorem 1.1 for pair correlation with density w_D; the geodesic–root dictionary is Theorem 5.1 and related formulas.