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

In this paper, we show how quantum modular forms naturally arise in the ergodic theory of circle rotations. Working with the classical Birkhoff sum $S_N(\alpha )={\sum }_{n=1}^N(\{n\alpha \}-1\mathrm{/}2)$, 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 $\alpha \in [0,1]$.