On the order of magnitude of Sudler products

Abstract

Given an irrational number α ∈ (0, 1), the Sudler product is defined by P N (α) = N r=1 2| sin πrα|. 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 lim sup N →∞ P N (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 characterization of the values of a for which lim inf N →∞ P N (α) > 0 and lim sup N →∞ P N (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 named author, Larcher, Pillichshammer, Saad Eddin, and Tichy.

On extreme values for the Sudler product of quadratic irrationals

“…Later Aistleitner, Technau and Zafeiropoulos [3] found a very close connection between the behaviour of lim inf N →∞ P N (α) and lim sup…”

“…Note that the Fibonacci numbers are the denominators of the continued fraction convergents of φ, hinting at a connection between the Sudler product of α and its Diophantine approximation properties. This was established more broadly by Aistleitner, Technau and Zafeiropoulos [3] who generalized Mestel and Verschueren's work to quadratic irrationals of the form β(b) = [0; b, b, . .…”

“…It was already shown in [3] that lim n→∞ P qn (β) = C b , although the closed expression differs from (4). The expression used in Theorem 1 first appeared in a more general setting in [2] where arbitrary quadratic irrationals were considered.…”

“…Both statements (i) and (ii) of Theorem 2 are optimal for quadratic irrationals with period length 1 in the sense that they cannot be extended to β(b) with b 6: By Theorem A, we know that in that case lim inf N →∞ P N (β) = 0, already making (7) impossible to hold. Since Aistleitner, Technau and Zafeiropoulus showed in [3] that for any b ∈ N we have lim n→∞ P qn (β) = C b > 0, also P qn (β) P N (β) must be violated for any b 6 and some n, N sufficiently large. By a similar argument, we see that P N (φ) P qn−1 (β) eventually fails as well.…”

“…Shifted Sudler products. We follow a decomposition approach that was implicitly used already in [18] to prove lim inf N →∞ P N (φ) > 0 and more explicitly, in later literature [1,2,3]. Here, the Sudler product P N (β) is decomposed into a finite product with factors of the form…”