Fine scale distribution of roots of quadratic congruences
We consider solutions (roots) x mod m of the quadratic congruence x^2 = D mod m for a fixed, squarefree integer D. Besides these roots being a classical object of study, statistical information on their distribution can be crucial input into the methods of analytic number theory, as seen in works by Iwaniec, for example. In joint work with Jens Marklof, we study the fine-scale distribution of these roots by seeing them as return times of the horocycle flow for a specific section in SL(2,Z)\SL(2,R), analogous to Athreya-Cheung's interpretation of the Boca-Cobeli-Zaharescu map for Farey fractions.
