论文标题
苏德勒产品价值分布的度量密度结果
Metric density results for the value distribution of Sudler products
论文作者
论文摘要
我们研究了Sudler产品$ p_n（α）的价值分布：= \ prod_ {n = 1}^{n} \ lvert2 \ sin（πnα）\ rvert $ for Lebesgue-几乎所有不合理$α$。我们表明，对于每个非偿还函数$ψ：（0，\ infty）\ to（0，\ infty）$，带有$ \ sum_ {k = 1}^{\ infty} \ frac {\ frac {1} {ψ（ψ（ψ（k）} = \ infty $ \ leq -月（\ log n）\} $具有上限$ 1 $，它回答了Bence Borda的问题。另一方面，我们证明$ \ {n \ in \ mathbb {n}：\ log p_n（α）\ geqψ（\ log n）\} $具有上部密度至少$ \\ frac {1} {1} {2} {2} {2 \ geq c $对于某些足够大的$ c> 0 $。
We study the value distribution of the Sudler product $P_N(α) := \prod_{n=1}^{N}\lvert2\sin(πn α)\rvert$ for Lebesgue-almost every irrational $α$. We show that for every non-decreasing function $ψ: (0,\infty) \to (0,\infty)$ with $\sum_{k=1}^{\infty} \frac{1}{ψ(k)} = \infty$, the set $\{N \in \mathbb{N}: \log P_N(α) \leq -ψ(\log N)\}$ has upper density $1$, which answers a question of Bence Borda. On the other hand, we prove that $\{N \in \mathbb{N}: \log P_N(α) \geq ψ(\log N)\}$ has upper density at least $\frac{1}{2}$, with remarkable equality if $\liminf_{k \to \infty} ψ(k)/(k \log k) \geq C$ for some sufficiently large $C > 0$.
