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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:433

Article

Verschueren, Paul¹  Mestel, Ben¹ 

[1] Open Univ, Dept Math & Stat, Milton Keynes MK7 6AA, Bucks, England

关键词: Asymptotic growth;    Renormalisation;    Self-similarity;    Sine product;   

DOI  :  10.1016/j.jmaa.2015.06.014

来源: Elsevier

【 摘 要 】

We study the growth at the golden rotation number omega = (root 5 - 1)/2 of the function sequence P-n (omega) = Pi(n)(r=1) vertical bar 2sin pi r omega vertical bar. 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 sub-sequence Q(n), = vertical bar Pi(Fn)(r=1) 2 sin pi r omega vertical bar 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 (2011) [10]. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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