Intersecting Geodesics on the Modular Surface

The paper proves that the intersection points and angles between a fixed compact geodesic segment β ⊂ {y < T} and the union Cd of closed geodesics of discriminant d equidistribute with main term (6/π^2) l(β) l(Cd) ∫_{θ1}^{θ2} sin θ dθ and error Oε(d^{-25/3584+ε}), uniformly for θ2−θ1, l(β) ≳ d^{-25/7168} (Theorem 1.1; see 2101.08768.pdf) . The proof uses the modular intersection kernel, smooth majorant/minorant approximations for the discontinuous angular window, and an effective Duke-type equidistribution for the lifted closed geodesics (Theorem 1.5) with power-saving Oε(d^{-25/512+ε}) against compactly supported smooth test functions supported in {y<T} , plus explicit kernel integrals and Sobolev control (Lemma 5.3, Corollary 5.4) . The candidate solution follows the same architecture: it uses the modular intersection kernel, smoothing, and effective equidistribution to get the stated main term and error, and it enforces the same small-scale thresholds. Two small inaccuracies are present: it (i) states an “exact identity” for the counting functional before smoothing (the paper instead inserts smooth majorant/minorant to bound the discontinuous gate), and (ii) it quotes the equidistribution saving as d^{-25/3584+ε} for test functions, whereas the paper’s test-function equidistribution saving is stronger, d^{-25/512+ε}, which later degrades to d^{-25/3584+ε} after the smoothing choices used in the intersection problem. These do not affect the final conclusion.