Continuity and value distribution of quantum modular forms

Quantum modular forms are functions $f$ defined on the rationals whose period functions, such as $\psi(x):= f(x) - x^{-k} f(-1/x)$ (for level 1), satisfy some continuity properties. In the case of $k=0$, $f$ can be interpreted as a Birkhoff sum associated with the Gauss map. In particular, under mild hypotheses on $\psi$, one can show convergence to a stable law. If $k$ is non-zero, the situation is rather different and we can show that mild conditions on $\psi$ imply that $f$ itself has to exhibit some continuity property. Finally, we discuss the convergence in distribution also in this case. This is a joint work with Sary Drappeau.