The Size of (q; q)nforqon the Unit Circle

Abstract

There is increasing interest in q-series with |q| =1. In analysis of these, all important role is played by the behaviour as n Ä ofWe show, for example, that for almost all q on the unit circle log |(q; q) n |=O(log n) 1+=iff =>0. Moreover, if q=exp(2?i{) where the continued fraction of { has bounded partial quotients, then the above relation is valid with ==0. This provides an interesting contrast to the well known geometric growth as n Ä of1999 Academic Press

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

“…for any ε > 0, and this holds for almost every α [8]. In the opposite direction, P n (α) grows almost linearly for infinitely many n. We have that lim sup n→∞ log P n (α) log n ≥ 1 for all irrationals α ∈ (0, 1).…”

“…In his 1999 paper [8], Lubinsky illustrates a significant difference in nature of P n (α) depending on whether or not the continued fraction expansion of α has bounded coefficients. If this is the case, then there exist positive constants C 1 and C 2 such that…”

“…Theorem 5.1 is a consequence of a result by Lubinsky (Proposition 5.2 below). Lubinsky himself claims in [8] that Theorem 5.1 is true for a general threshold K independent of M . However, this is not rigorously proven, and we have not managed to verify it.…”

Asymptotic behaviour of the Sudler product of sines for quadratic irrationals

“…As a consequence of Theorem 1.1, Mestel and Verschueren show in [15] that one can establish polynomial bounds on P n (α) when α = ω is the golden mean. Specifically, they show that in this case there exist constants K 1 ≤ 0 < 1 ≤ K 2 such that n K 1 ≤ P n (α) ≤ n K 2 , (8.1) for all n ∈ N. It is worth mentioning that this is not a new result; in a paper from 1999, Lubinsky studies the product P n (α) in the language of q-series [10]. In particular, he proves that (8.1) holds whenever α = [0; a 1 , a 2 , .…”

On Weyl products and uniform distribution modulo one

“…Acknowledgment. We thank Dmitriy Bilyk who drew our attention to the results given in [23,30], and [41].…”