On extreme values for the Sudler product of quadratic irrationals

Abstract

Given a real number $\alpha$ and a natural number $N$, the Sudler product is defined by $P_N(\alpha) = \prod_{r=1}^{N} 2 \left\lvert \sin(\pi\left(r\alpha \right))\right\rvert.$ 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 $\min_{N \geq 1} P_N(\phi) = P_1(\phi)$, thereby proving a conjecture of Grepstad, Kaltenb\"ock and Neum\"uller. Furthermore, we find closed expressions for $\liminf_{N \to \infty} P_N(\phi)$ and $\limsup_{N \to \infty} \frac{P_N(\phi)}{N}$ whose numerical values can be approximated arbitrarily well. We generalize these results to the case of quadratic irrationals $\beta$ with continued fraction expansion $\beta = [0;b,b,b\ldots]$ where $1 \leq b \leq 5$, completing the calculation of $\liminf_{N \to \infty} P_N(\beta)$, $\limsup_{N \to \infty} \frac{P_N(\beta)}{N}$ for $\beta$ being an arbitrary quadratic irrational with continued fraction expansion of period length 1.