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 $P_N(\alpha )={\prod }_{r=1}^{N}2\left|\sin (\pi \left(r\alpha \right))\right|.$ Denoting by $F_n$ the $n$--th Fibonacci number and by $\varphi$ the Golden Ratio, we show that for $F_{n-1}\le N<F_n$, we have $P_{F_{n-1}}(\varphi )\le P_N(\varphi )\le P_{F_n-1}(\varphi )$ and ${\min }_{N\ge 1}{P}_N(\varphi )=P_1(\varphi )$, thereby proving a conjecture of Grepstad, Kaltenböck and Neumüller. Furthermore, we find closed expressions for ${\mathrm{liminf}}_{N\to \infty }{P}_N(\varphi )$ and ${\mathrm{limsup}}_{N\to \infty }\frac{P_N(\varphi )}{N}$ whose numerical values can be approximated arbitrarily well. We generalize these results to the case of quadratic irrationals $\beta$ with continued fraction expansion $\beta =[0;b,b,b\dots ]$ where $1\le b\le 5$, completing the calculation of ${\mathrm{liminf}}_{N\to \infty }{P}_N(\beta )$, ${\mathrm{limsup}}_{N\to \infty }\frac{P_N(\beta )}{N}$ for $\beta$ being an arbitrary quadratic irrational with continued fraction expansion of period length 1.

arXiv comment37 pages, 6 figures. Comments are welcome