论文标题
关于Sudler产品的数量级
On the order of magnitude of Sudler products
论文作者
论文摘要
给定一个不合理的数字$α\在（0,1）$中，Sudler产品由$ p_n（α）= \ prod_ {r = 1}^{n} 2 | \sinπrα| $定义。回答Grepstad，Kaltenböck和Neumüller的问题，当$α$是黄金比率$（\ sqrt {5} +1）/2 $时，我们证明了扭曲的Sudler产品的渐近公式，并确定在这种情况下，在这种情况下，$ \ limsup_ to $ \ limsup_ to \ to \ to \ f \ feftty $ nfty $ n <对于某些整数$ a \ geq 1 $，我们获得了二次非理性$α$的二次非理性$α$，并为某些整数$ a \ geq 1 $ $α= $α= $ a $ a $ a $ a $ \ liminf_ { p_n（α） / n <\ infty $保留。我们确定有一个（尖锐的）过渡点$ a = 6 $，并作为副产品解决了第一个命名作者，拉尔切尔，皮利希·史瑟，萨德·埃丁和蒂奇的问题。
Given an irrational number $α\in(0,1)$, the Sudler product is defined by $P_N(α) = \prod_{r=1}^{N}2|\sinπrα|$. Answering a question of Grepstad, Kaltenböck and Neumüller we prove an asymptotic formula for distorted Sudler products when $α$ is the golden ratio $(\sqrt{5}+1)/2$ and establish that in this case $\limsup_{N \to \infty} P_N(α)/N < \infty$. We obtain similar results for quadratic irrationals $α$ with continued fraction expansion $α= [a,a,a,\dots]$ for some integer $a \geq 1$, and give a full characterization of the values of $a$ for which $\liminf_{N \to \infty} P_N(α)>0$ and $\limsup_{N \to \infty} P_N(α) / N < \infty$ hold, respectively. We establish that there is a (sharp) transition point at $a=6$, and resolve as a by-product a problem of the first named author, Larcher, Pillichshammer, Saad Eddin, and Tichy.
