  We study the value distribution of the Sudler product $\prod_{n=1}^N |2 \sin
(\pi n \alpha )|$ and the Diophantine product $\prod_{n=1}^N (2e\| n \alpha
\|)$ for various irrational $\alpha$, as $N$ ranges in a long interval of
integers. At badly approximable irrationals these products exhibit strong
concentration around $N^{1/2}$, and at quadratic irrationals they even satisfy
a central limit theorem. In contrast, at almost every $\alpha$ we observe an
interesting anticoncentration phenomenon when the typical and the extreme
values are of the same order of magnitude. Our methods are equally suited for
the value distribution of Birkhoff sums $\sum_{n=1}^N f(n \alpha )$ for circle
rotations. Using Diophantine approximation and Fourier analysis, we find the
first and second moment for an arbitrary periodic $f$ of bounded variation, and
(almost) prove a conjecture of Bromberg and Ulcigrai on the appropriate scaling
factor in a so-called temporal limit theorem.
