Asymptotic behaviour of the Sudler product of sines for quadratic irrationals

We study the asymptotic behaviour of the sequence of sine products Pn(α)=∏r=1n|2sin⁡πrα| for real quadratic irrationals α. In particular, we study the subsequence Qn(α)=∏r=1qn|2sin⁡πrα|, where qn is the nth best approximation denominator of α, and show that this subsequence converges to a periodic sequence whose period equals that of the continued fraction expansion of α. This verifies a conjecture recently posed by Verschueren and Mestel (2016) [15].