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

On the metric upper density of Birkhoff sums for irrational rotations