On extreme values for the Sudler product of quadratic irrationals

TL;DR

This paper investigates the behavior of the Sudler product for quadratic irrationals, proving bounds, limits, and extending results to a class of quadratic irrationals with specific continued fraction expansions.

Contribution

It proves a conjecture about the Sudler product for the golden ratio and extends the analysis to quadratic irrationals with periodic continued fractions.

Findings

Established bounds for the Sudler product at Fibonacci indices.

Derived explicit formulas for the lim inf and lim sup of the product.

Extended results to quadratic irrationals with period-1 continued fractions.

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.