On extreme values for the Sudler product of quadratic irrationals

2021/11/25 by Manuel Hauke, Hauke, Manuel · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2111.12974

openalex publication_date 2021/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a real number α and a natural number N, the Sudler product is defined by P_N(α) = ∏_r=1^N 2 | sin(π(rα))|. Denoting by F_n the n--th Fibonacci number and by ϕ the Golden Ratio, we show that for F_n-1 ≤ N < F_n, we have P_F_n-1(ϕ)≤ P_N(ϕ) ≤ P_F_n-1(ϕ) and min_{N ≥ 1} P_N(ϕ) = P₁(ϕ), thereby proving a conjecture of Grepstad, Kaltenböck and Neumüller. Furthermore, we find closed expressions for \liminf_{N → ∞} P_N(ϕ) and \limsup_{N → ∞} (P_N(ϕ))/(N) whose numerical values can be approximated arbitrarily well. We generalize these results to the case of quadratic irrationals β with continued fraction expansion β= [0;b,b,b…] where 1 ≤ b ≤ 5, completing the calculation of \liminf_{N → ∞} P_N(β), \limsup_{N → ∞} (P_N(β))/(N) for β being an arbitrary quadratic irrational with continued fraction expansion of period length 1.

Cited by

• On the behavior of the colored Jones polynomial of the figure-eight knot under modular transformations

Related

• On the order of magnitude of Sudler products
• On the order of magnitude of Sudler products II
• On the asymptotic behaviour of Sudler products for badly approximable numbers
• Asymptotic behaviour of the Sudler product of sines for quadratic\n irrationals
• Hausdorff dimension estimates for Sudler products with positive lower bound