@phdthesis{discovery10203381,
          school = {UCL (University College London)},
            year = {2025},
           title = {Distributional Results for Geodesic Segments in the Hyperbolic Plane},
            note = {Copyright {\copyright} The Author 2025. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author's request.},
           pages = {1--136},
           month = {January},
        abstract = {For {\ensuremath{\Gamma}} a cofinite Fuchsian group, and l a fixed closed geodesic, we study refined
asymptotics of the number of images of l over {\ensuremath{\Gamma}} that have a distance from l less than or equal to X. In particular, we partition the images into four cases, according to orientation, and prove that they all contribute asymptotically one-fourth of the total. This was originally studied by A. Good. For {\ensuremath{\Gamma}}\_1 the stabilizer of l, this is equivalent to counting the number of double cosets in {\ensuremath{\Gamma}}\_1{$\backslash$}{\ensuremath{\Gamma}}/{\ensuremath{\Gamma}}\_1 with prescribed signs for its entries, according to a certain growth parameter. We achieve this by developing new modified relative trace formulae, as well as bounds for hyperbolic periods in mean square.  We give a new concrete proof of the error bound O(X{\^{ }}(2/3)) that appears in the works of Good and Hejhal. Furthermore, we prove a new bound O(X{\^{ }}(1/2) logX) for the mean square of the error. To that end, we obtain large sieve inequalities
with weights the hyperbolic periods of Maa{\ss} forms of even weight. This is inspired by work of Chamizo, who proved a large sieve inequality with weights the values of Maa{\ss} forms of weight 0. We also prove ? results, supporting the conjectural best error term O\_{\ensuremath{\epsilon}}(X{\^{ }}(1/2+{\ensuremath{\epsilon}}))\$. For particular arithmetic groups, we provide interpretations in terms of correlation sums of the number of ideals of norm at most X in associated number fields, generalizing previous examples due to Hejhal.},
             url = {https://discovery.ucl.ac.uk/id/eprint/10203381/},
          author = {Voskou, Marios}
}