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Abstract
Given a real number $\alpha$ and a natural number $N$ , the Sudler product is defined by $P_N(\alpha) = \prod_{r=1}^{N} 2 \sin(\pi (r\alpha))$. Denoting by $F_n$ the $n$–th Fibonacci number and by $\phi$ the
Golden Ratio, we show that for $F_{n−1} \leq N < F_n$ , we have $P_{F_{n−1}}(\phi) \leq P_N(\phi) \leq P_{F_n −1 }(\phi)$ and
$min_{N \geq 1} P_N (\varphi) = P_1(\phi)$, thereby proving a conjecture of Grepstad, Kaltenböck and Neumüller.
Furthermore, we find closed expressions for $\liminf_{N \to \infty} P_N(\phi)$ and $\limsup_{N \to \infty} P_N(\phi)/N$ whose
numerical values can be approximated arbitrarily well. We generalize these results to the case
of quadratic irrationals $\beta$ with continued fraction expansion $\beta = [0; b, b, b,. . .]$ where $1 \leq b \leq 5$,
completing the calculation of $\liminf_{N \to \infty} P_N(\beta), \limsup{N \to \infty} P_N(\beta)/N for $\beta$ being an arbitrary
quadratic irrational with continued fraction expansion of period length 1.
Golden Ratio, we show that for $F_{n−1} \leq N < F_n$ , we have $P_{F_{n−1}}(\phi) \leq P_N(\phi) \leq P_{F_n −1 }(\phi)$ and
$min_{N \geq 1} P_N (\varphi) = P_1(\phi)$, thereby proving a conjecture of Grepstad, Kaltenböck and Neumüller.
Furthermore, we find closed expressions for $\liminf_{N \to \infty} P_N(\phi)$ and $\limsup_{N \to \infty} P_N(\phi)/N$ whose
numerical values can be approximated arbitrarily well. We generalize these results to the case
of quadratic irrationals $\beta$ with continued fraction expansion $\beta = [0; b, b, b,. . .]$ where $1 \leq b \leq 5$,
completing the calculation of $\liminf_{N \to \infty} P_N(\beta), \limsup{N \to \infty} P_N(\beta)/N for $\beta$ being an arbitrary
quadratic irrational with continued fraction expansion of period length 1.
Original language  English 

Pages (fromto)  4182 
Number of pages  43 
Journal  Acta Arithmetica 
Volume  204 
DOIs  
Publication status  Published  2022 
Keywords
 Diophantine approximation
 Continued fractions
 Trigonometric product
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The asymptotic behaviour of Sudler products
Manuel Hauke (Speaker)
29 Nov 2022Activity: Talk or presentation › Talk at workshop, seminar or course › Science to science