On the order of magnitude of Sudler products II

Grepstad, Sigrid; Neumüller, Mario; Zafeiropoulos, Agamemnon

doi:10.48550/arxiv.2109.04342

Cited by 4 publications

(11 citation statements)

References 7 publications

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“…On the other hand, Grepstad, Kaltenböck and Neumüller showed in [13] that lim inf N →∞ P N (φ) > 0 for φ being the Golden Ratio, answering the question negatively. This counterexample was extended in [4,15] to certain quadratic irrationals that have only particularly small partial quotients.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Metric density results for the value distribution of Sudler products

Manuel¹

2022

Preprint

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We study the value distribution of the Sudler product P N (α) := N n=1 |2 sin(πnα)| for Lebesgue-almost every irrational α. We show that for every non-decreasing functionupper density 1, which answers a question of Bence Borda. On the other hand, we prove that {N ∈ N : log P N (α) ≥ ψ(log N )} has upper density at least 1 2 , with remarkable equality if lim inf k→∞ ψ(k)/(k log k) ≥ C for some sufficiently large C > 0.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Metric density results for the value distribution of Sudler products

Manuel¹

2022

Preprint

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“…Directions for further research. Recently, Grepstad, Neumüller and Zafeiropoulos [17] studied the behaviour of lim inf N →∞ P N (α), lim sup N →∞…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On extreme values for the Sudler product of quadratic irrationals

Manuel¹

2021

Preprint

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Given a real number α and a natural number N , the Sudler product is defined by P N (α) = N r=1 2 sin(π (rα)) . Denoting by F n the n-th Fibonacci number and by φ the Golden Ratio, we show that for F n−1 N < F n , we have P Fn−1 (φ) P N (φ) P Fn−1 (φ) and min N 1 P N (φ) = P 1 (φ), thereby proving a conjecture of Grepstad, Kaltenböck and Neumüller. Furthermore, we find closed expressions for lim inf N →∞ P N (φ) and lim sup N →∞ P N (φ) N whose numerical values can be approximated arbitrarily well. We generalize these results to the case of quadratic irrationals β with continued fraction expansion β = [0; b, b, b . . .] where 1 b 5, completing the calculation of lim inf N →∞ P N (β), lim sup N →∞ P N (β) Nfor β being an arbitrary quadratic irrational with continued fraction expansion of period length 1.

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“…This is called Sudler product, and has a long history going back at least to a paper of Erdős and Szekeres [12]. See [1,2,3,14,15,16,17,24] for recent papers concerned with the behavior of such products. Note that…”

Section: Shifted Trigonometric Productsmentioning

confidence: 99%

“…provided that b k (N ) ≥ 1; an obviously modified formula holds when b k (N ) = 0. According to the porism mentioned after (17), the assumption a k+1 ≥ 7 ensures that…”

Section: The Local 5/6-principlementioning

confidence: 99%

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A conjecture of Zagier and the value distribution of quantum modular forms

Aistleitner¹,

Borda²

2021

Preprint

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In his influential paper on quantum modular forms, Zagier developed a conjectural framework describing the behavior of certain quantum knot invariants under the action of the modular group on their argument. More precisely, when J K,0 denotes the colored Jones polynomial of a knot K, Zagier's modularity conjecture describes the asymptotics of the quotient J K,0 (e 2πiγ(x) )/J K,0 (e 2πix ) as x → ∞ along rationals with bounded denominators, where γ ∈ SL(2, Z). This problem is most accessible for the figure-eight knot 4 1 , when the colored Jones polynomial has an explicit expression in terms of the q-Pochhammer symbol. Zagier also conjectured that the function h(x) = log(J 41,0 (e 2πix )/J 41,0 (e 2πi/x )) can be extended to a function on R which is continuous at irrationals. In the present paper, we prove Zagier's continuity conjecture for all irrationals which have unbounded partial quotients in their continued fraction expansion. In particular, the continuity conjecture holds almost everywhere on the real line. We also establish a smooth approximation of h, uniform over all rationals, in accordance with the modularity conjecture. As an application, we find the limit distribution (after a suitable centering and rescaling) of log(J 41,0 (e 2πix )), when x ranges over all reduced rationals in (0, 1) with denominator at most N , as N → ∞, thereby confirming a conjecture of Bettin and Drappeau.

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On the order of magnitude of Sudler products II

Cited by 4 publications

References 7 publications

Metric density results for the value distribution of Sudler products

Metric density results for the value distribution of Sudler products

On extreme values for the Sudler product of quadratic irrationals

A conjecture of Zagier and the value distribution of quantum modular forms

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