  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 certain 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. Birkhoff sums
also satisfy a central limit theorem at certain quadratic irrationals.
