Asymptotic behaviour of the Sudler product of sines for quadratic irrationals

الأصل · EN

We study the asymptotic behaviour of the sequence of sine products Pₙ(α) = ∏ᵣ₌₁ⁿ |2 πr α| for real quadratic irrationals α. In particular, we study the subsequence Qₙ(α)=∏ᵣ₌₁qⁿ |2 πr α|, 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 Mestel and Verschueren.