论文标题
关于苏德勒产品的渐近行为，数字差不多
On the asymptotic behaviour of Sudler products for badly approximable numbers
论文作者
论文摘要
给定一个差不多的数字$α$，我们研究了由$ p_n（α）= \ prod_ {r = 1}^n 2 |定义的Sudler产品的渐近行为\sinπrα| $。我们表明$ \ liminf_ {n \ to \ infty} p_n（α）= 0 $和$ \ limsup_ {n \ to \ infty} p_n（α）/n = \ infty $，每当持续分数扩展$α$ lim -α$ 7 $ 7 $ 7时，每当部分分数扩展$ 7 $ 7时。这改善了卢宾斯基（Lubinsky）为一般情况获得的结果，以及Grepstad，Neumüller和Zafeiropoulos对于二次非理性的特殊情况。此外，我们证明，即使将$α$限制为二次非理性，该门槛值$ 7 $也是最佳的，这给了后者作者的问题带来负面答案。
Given a badly approximable number $α$, we study the asymptotic behaviour of the Sudler product defined by $P_N(α) = \prod_{r=1}^N 2 | \sin πr α|$. We show that $\liminf_{N \to \infty} P_N(α) = 0$ and $\limsup_{N \to \infty} P_N(α)/N = \infty$ whenever the sequence of partial quotients in the continued fraction expansion of $α$ exceeds $7$ infinitely often. This improves results obtained by Lubinsky for the general case, and by Grepstad, Neumüller and Zafeiropoulos for the special case of quadratic irrationals. Furthermore, we prove that this threshold value $7$ is optimal, even when restricting $α$ to be a quadratic irrational, which gives a negative answer to a question of the latter authors.
