On the growth of Sudler's sine product at the golden rotation number

Abstract

We study the growth at the golden rotation number of Sudler's sine product. 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, and prove that the renormalisation subsequence converges to a constant. From this we show rigorously that the growth 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).