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.

DOI: https://doi.org/10.1016/j.jmaa.2015.06.014

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).

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