On the metric upper density of Birkhoff sums for irrational rotations

This article examines the value distribution of (Formula presented) for almost every α where N ∈ 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 , α ) 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.