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

Abstract

Abstract. We study the growth at the golden rotation number ω = ( √ 5 − 1)/2 of the function sequence P n (ω) = n r=1 |2 sin πrω|. This sequence has been variously studied elsewhere as a skew product of sines, Birkho 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 = Fn r=1 2 sin πrω 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 (ω) 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 Birkho sums for the golden rotation. Nonlinearity, 24(11):31153127, 2011).

5. On the asymptotic behavior of the sine productΠn r =1 /2 sin πrα/

“…First of all, should the upper bound in (4.2) hold, then it would follow that the growth of P n (ϕ) is at most linear. Using the convergence of the subsequence P Fn (ϕ), it is derived in [12] that P Fn−1 (ϕ) ≤ cF n , and combining this with (4.2) we get P N (ϕ) ≤ cF n ≤ 2cN.…”

“…Our interest in these questions was sparked by a recent paper by Mestel and Verschueren [12], where the special case α = ( √ 5 − 1)/2 is studied in great detail. We review key results from this paper in Section 3.…”

Asymptotic behaviour of the Sudler product of sines for quadratic irrationals

“…|2 sin πrα|, (1.1) where (q n ) n≥0 are the best approximation denominators of α. In a recent paper, Mestel and Verschueren study Q n (α) in the special case where α = ω := ( √ 5 − 1)/2 is the fractional part of the golden mean [15]. For this case, it was suggested by Knill and Tangerman in [7] that the limit value lim n→∞ Q n (ω) might exist, and this is confirmed by Mestel and Verschueren.…”

“…• Following Mestel and Verschueren [15], we introduce a generalized sum and product notation: given a summable sequence (b r ) r∈N , we define the step function f (t) = b r for t ∈ [r, r + 1). Then for any x, y ∈ R where x ≤ y, we let This allows us to define sums and products with real, rather than just integer, upper and lower bounds.…”

“…It appears that research has been carried out simultaneously, and partly independently, in different mathematical disciplines, resulting in a number of different terminologies and representations of P n (α). For a brief summary of key results we recommend the introduction of [15].…”