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

TL;DR

This paper analyzes the growth behavior of Sudler's sine product at the golden rotation number, establishing bounds and convergence properties that explain observed self-similar growth patterns.

Contribution

It provides a rigorous mathematical analysis of Sudler's sine product growth at the golden rotation, including convergence of a renormalization subsequence and power law bounds.

Findings

Renormalization subsequence converges to a constant.

Growth is bounded by power laws.

Supports previous experimental observations.

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).