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

Abstract

We study the growth at the golden rotation number ω=(5−1)/2 of the function sequence Pn(ω)=∏r=1n|2sin⁡πrω|. 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 Pn, namely the sub-sequence Qn=|∏r=1Fn2sin⁡πrω| for Fibonacci numbers Fn, and prove that this renormalisation subsequence converges to a constant. From this we show rigorously that the growth of Pn(ω) is bounded by power laws. This provides the theoretical basis to explain recent experimental results reported by Knill and Tangerman (2011) [10].