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On the order of magnitude of Sudler products

Given an irrational number $α\in(0,1)$, the Sudler product is defined by $P_N(α) = \prod_{r=1}^{N}2|\sinπrα|$. Answering a question of Grepstad, Kaltenböck and Neumüller we prove an asymptotic formula for distorted Sudler products when $α$ is the golden ratio $(\sqrt{5}+1)/2$ and establish that in this case $\limsup_{N \to \infty} P_N(α)/N < \infty$. We obtain similar results for quadratic irrationals $α$ with continued fraction expansion $α= [a,a,a,\dots]$ for some integer $a \geq 1$, and give a full characterization of the values of $a$ for which $\liminf_{N \to \infty} P_N(α)>0$ and $\limsup_{N \to \infty} P_N(α) / N < \infty$ 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.