On the asymptotic behavior of Sudler products along subsequences

2021/03/26 by Neumüller, Mario
#11J70 (primary) #11K31 (secondary) #11L15 #26D05 #41A60 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2103.14307

Abstract

Let α∈ (0,1) and irrational. We investigate the asymptotic behaviour of sequences of certain trigonometric products (Sudler products) (P_N(α))_N∈ℕ with P_N(α) =∏_r=1^N|2sin(πr α)|. More precisely, we are interested in the asymptotic behaviour of subsequences of the form (P_qn(α)(α))_n∈ℕ, where q_n(α) is the nth best approximation denominator of α. Interesting upper and lower bounds for the growth of these subsequences are given, and convergence results, obtained by Mestel and Verschueren (see arXiv:1411.2252math[DS]) and Grepstad and Neumüller (see arXiv:1801.09416[math.NT]), are generalized to the case of irrationals with bounded continued fraction coefficients.

Related

• Asymptotic behaviour of the Sudler product of sines for quadratic\n irrationals
• On the order of magnitude of Sudler products II
• On the asymptotic behaviour of Sudler products for badly approximable numbers
• On the order of magnitude of Sudler products
• On the asymptotic behaviour of the sine product ∏_r=1ⁿ|2sin(πr α)|