On the metric upper density of Birkhoff sums for irrational rotations

This article examines the value distribution of $S_{N}(f, \alpha) := \sum_{n=1}^N f(n\alpha)$ for almost every $\alpha$ where $N \in \mathbb{N}$ is ranging over a long interval and $f$ is a $1$-periodic function with discontinuities or logarithmic singularities at rational numbers. We show that for $N$ in a set of positive upper density, the order of $S_{N}(f, \alpha)$ is of Khintchine-type, unless the logarithmic singularity is symmetric. Additionally, we show the asymptotic sharpness of the Denjoy-Koksma inequality for such $f$, with applications in the theory of numerical integration. Our method also leads to a generalized form of the classical Borel-Bernstein Theorem that allows very general modularity conditions.