On extreme values for the Sudler product of quadratic irrationals

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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.
Original languageEnglish
Pages (from-to)41-82
Number of pages43
JournalActa Arithmetica
Volume204
DOIs
Publication statusPublished - 2022

Keywords

  • Diophantine approximation
  • Continued fractions
  • Trigonometric product

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