Abstract
We study the value distribution of the Sudler product $P_N(\alpha) := \prod_{n=1}^{N}\lvert2\sin(\pi n \alpha)\rvert$ for Lebesguealmost every irrational $\alpha$. We show that for every nondecreasing function \\\mbox{$\psi: (0,\infty) \to (0,\infty)$} with $\sum_{k=1}^{\infty} \frac{1}{\psi(k)} = \infty$, the
set $\{N \in \mathbb{N}: \log P_N(\alpha) \leq \psi(\log N)\}$ has upper density $1$, which answers a question of Bence Borda.
On the other hand, we prove that $\{N \in \mathbb{N}: \log P_N(\alpha) \geq \psi(\log N)\}$ has upper density at least $\frac{1}{2}$, with remarkable equality if $\liminf_{k \to \infty} \psi(k)/(k \log k) \geq C$ for some sufficiently large $C > 0$.
\end{abstract}
set $\{N \in \mathbb{N}: \log P_N(\alpha) \leq \psi(\log N)\}$ has upper density $1$, which answers a question of Bence Borda.
On the other hand, we prove that $\{N \in \mathbb{N}: \log P_N(\alpha) \geq \psi(\log N)\}$ has upper density at least $\frac{1}{2}$, with remarkable equality if $\liminf_{k \to \infty} \psi(k)/(k \log k) \geq C$ for some sufficiently large $C > 0$.
\end{abstract}
Original language  English 

Number of pages  13 
DOIs  
Publication status  Accepted/In press  28 Sep 2022 
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Best Paper Award der Doctoral School Mathematics and Scientific Computing
Hauke, Manuel (Recipient), 18 Nov 2022
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