TY - JOUR
T1 - On the order of magnitude of Sudler products
AU - Aistleitner, Christoph
AU - Technau, Niclas
AU - Zafeiropoulos, Agamemnon
N1 - Publisher Copyright:
. © 2023 by Johns Hopkins University Press.
PY - 2023/6/1
Y1 - 2023/6/1
N2 - Given an irrational number α ∈ (0,1), the Sudler product is defined by PN (α) = ∏Nr=1 2|sinπrα|. Answering a question of Grepstad, Kaltenböck and Neumüller we prove an asymptotic formula for distorted Sudler products when α is the golden ratio (√5 +1)/2 and establish that in this case limsupN→∞ PN (α)/N < ∞. We obtain similar results for quadratic irrationals α with continued fraction expansion α = [a,a,a,…] for some integer a ≥ 1, and give a full character-isation of the values of a for which liminfN→∞ PN (α) > 0 and limsupN→∞ PN (α)/N < ∞ hold, respectively. We establish that there is a (sharp) transition point at a = 6, and resolve as a by-product a problem of the first author, Larcher, Pillichshammer, Saad Eddin, and Tichy.
AB - Given an irrational number α ∈ (0,1), the Sudler product is defined by PN (α) = ∏Nr=1 2|sinπrα|. Answering a question of Grepstad, Kaltenböck and Neumüller we prove an asymptotic formula for distorted Sudler products when α is the golden ratio (√5 +1)/2 and establish that in this case limsupN→∞ PN (α)/N < ∞. We obtain similar results for quadratic irrationals α with continued fraction expansion α = [a,a,a,…] for some integer a ≥ 1, and give a full character-isation of the values of a for which liminfN→∞ PN (α) > 0 and limsupN→∞ PN (α)/N < ∞ hold, respectively. We establish that there is a (sharp) transition point at a = 6, and resolve as a by-product a problem of the first author, Larcher, Pillichshammer, Saad Eddin, and Tichy.
UR - http://www.scopus.com/inward/record.url?scp=85161504139&partnerID=8YFLogxK
U2 - 10.1353/ajm.2023.a897495
DO - 10.1353/ajm.2023.a897495
M3 - Article
AN - SCOPUS:85161504139
SN - 0002-9327
VL - 145
SP - 721
EP - 764
JO - American Journal of Mathematics
JF - American Journal of Mathematics
IS - 3
ER -