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Growth of the Sudler product of sines at the golden rotation number

Verschueren, Paul and Mestel, Ben (2016). Growth of the Sudler product of sines at the golden rotation number. Journal of Mathematical Analysis and Applications, 433(1) pp. 220–226.

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DOI (Digital Object Identifier) Link: https://doi.org/10.1016/j.jmaa.2015.06.014
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Abstract

We study the growth at the golden rotation number $\omega=(\sqrt{5}-1)/2$ of the function sequence $P_{n}(\omega)=\prod_{r=1}^{n}|2\sin\pi r\omega|$. This sequence has been variously studied elsewhere as a skew product of sines, Birkhoff sum, q-Pochhammer symbol (on the unit circle), and restricted Euler function. In particular we study the Fibonacci decimation of the sequence $P_{n}$, namely the subsequence $Q_{n}=\left|\prod_{r=1}^{F_{n}}2\sin\pi r\omega\right|$ for Fibonacci numbers $F_{n}$, and prove that this renormalisation subsequence converges to a constant. From this we show rigorously that the growth of $P_{n}(\omega)$ is bounded by power laws. This provides the theoretical basis to explain recent experimental results reported by Knill and Tangerman (Self-similarity and growth in Birkhoff sums for the golden rotation. Nonlinearity, 24(11):3115-3127,
2011).

Item Type: Journal Item
ISSN: 0022-247X
Extra Information: 2015 is an estimated publication date.
Keywords: Asymptotic growth, renormalisation, self-similarity, sine product
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 43460
Depositing User: Benjamin Mestel
Date Deposited: 12 Jun 2015 09:08
Last Modified: 30 Apr 2019 22:58
URI: http://oro.open.ac.uk/id/eprint/43460
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