Limit laws of maximal Birkhoff sums for circle rotations via quantum modular forms

Abstract: In this paper, we show how quantum modular forms naturally arise in the ergodic theory of circle rotations. Working with the classical Birkhoff sum $S_N(\alpha)=\sum_{n=1}^N ({ n \alpha }-1/2)$, we prove that the maximum and the minimum as well as certain exponential moments of $S_N(r)$ as functions of $r \in \mathbb{Q}$ satisfy a direct analogue of Zagier's continuity conjecture, originally stated for a quantum invariant of the figure-eight knot. As a corollary, we find the limit distribution of $\max_{0 \le N<M} S_N(\alpha)$ and $\min_{0 \le N<M} S_N(\alpha)$ with a random $\alpha \in [0,1]$.