On the order of magnitude of Sudler products

Given an irrational number $\alpha \in (0,1)$, the Sudler product is defined by $P_N(\alpha )={\prod }_{r=1}^{N}2\mathrm{\mid }\sin \pi r\alpha \mathrm{\mid }$. Answering a question of Grepstad, Kaltenb"ock and Neum"uller we prove an asymptotic formula for distorted Sudler products when $\alpha$ is the golden ratio $(\sqrt{5}+1)\mathrm{/}2$ and establish that in this case $\limsup_{N \to \infty} P_N(\alpha)/N < \infty$\mathrm{.}Weobtainsimilarresultsforquadraticirrationals$\alpha$ with continued fraction expansion $\alpha =[a,a,a,\dots ]$ for some integer $a \geq 1$,andgiveafullcharacterizationofthevaluesof$a$forwhich$\liminf_{N \to \infty} P_N(\alpha)>0$and$\limsup_{N \to \infty} P_N(\alpha) / N < \infty$ hold, respectively. We establish that there is a (sharp) transition point at $a=6$, and resolve as a by-product a problem of the first named author, Larcher, Pillichshammer, Saad Eddin, and Tichy.