Intersecting geodesics on the modular surface (Q6101527)

Let \(\mathbb{X}\) stand for the full modular surface \(\mathrm{PSL}_2(\mathbb{Z}) \setminus \mathbb{H}\). In the paper under the review, the authors introduce the term of the modular intersection kernel, which is then used to study intersections of geodesic curves on \(\mathbb{X}\). Let us fix a discriminant \(d\) and let \(C_d\) denote the union of closed geodesics with discriminant \(d\). Let \(\beta \subset \mathbb{X}\) stand for a compact geodesic segment and let \(p \in \beta \cap C_d\). We denote by \(\theta_p\) an angle of intersection between \(\beta\) and \(C_d\) at \(p\). Studying the introduced modular intersection kernel, the authors prove that the set \(\{ (p, \theta_p) : p \in \beta \cap C_d \}\) becomes equidistributed with respect to \(\sin \theta ds d \theta\) on \(\beta \times [0, \pi]\) with a power savings rate as \(d \rightarrow + \infty\). A similar result is also proven for the distribution of angles of intersections between \(C_{d_1}\) and \(C_{d_2}\) when \(d_1 + d_2 \rightarrow + \infty\).