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Abstract

For a real number $\alpha $ and a natural number $N$, the Sudler product is defined by $P_N(\alpha ) = \prod _{r=1}^{N} 2 \lvert \sin (\pi r\alpha )\rvert $. Denoting by $F_n$ the $n$th Fibonacci number and by $\phi $ the Golden Ratio, we show that for $F_{n-1} \le N \lt F_n$, we have $P_{F_{n-1}}(\phi )\le P_N(\phi ) \le P_{F_{n}-1}(\phi )$ and $\min _{N \ge 1} P_N(\phi ) = P_1(\phi )$, 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 $\beta = [0;b,b,\ldots ]$ where $1 \le b \le 5$, 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.