On the asymptotic behaviour of Sudler products for badly approximable numbers

Given a badly approximable number alpha, we study the asymptotic behaviour of the Sudler product defined by PN(alpha) = pi Nr=12|sinirr alpha|. We show that lim infN ->infinity PN (alpha) = 0 and lim supN ->infinity PN(alpha)/N = infinity whenever the sequence of partial quotients in the continued fraction expansion of alpha exceeds 7 infinitely often. This improves results obtained by Lubinsky for the general case, and by Grepstad, Neumuller and Zafeiropoulos for the special case of quadratic irrationals. Furthermore, we prove that this threshold value 7 is optimal, even when restricting alpha to be a quadratic irrational, which gives a negative answer to a question of the latter authors.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).