Metric density results for the value distribution of Sudler products

Manuel Hauke

doi:10.1090/proc/16269

Metric density results for the value distribution of Sudler products

Manuel Hauke^*

^*Corresponding author for this work

Institute of Analysis and Number Theory (5010)

Research output: Contribution to journal › Article › peer-review

Abstract

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 function ψ : (0, ∞) → (0, ∞) with^∞_k_{=1 ψ(}¹_k₎ = ∞, the set {N ∈ N: log P_N (α) ≤ −ψ(log N)} has upper 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 least¹₂ , with remarkable equality if lim inf_k→∞ ψ(k)/(k log k) ≥ C for some sufficiently large C > 0.

Original language	English
Pages (from-to)	2339-2351
Number of pages	13
Journal	Proceedings of the American Mathematical Society
Volume	151
Issue number	6
DOIs	https://doi.org/10.1090/proc/16269
Publication status	Published - 1 Jun 2023

Keywords

continued fraction
Diophantine approximation
metric number theory
Sudler product

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1090/proc/16269

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title = "Metric density results for the value distribution of Sudler products",

abstract = "We study the value distribution of the Sudler product PN (α):=Nn=1 |2 sin(πnα)| for Lebesgue-almost every irrational α. We show that for every non-decreasing function ψ : (0, ∞) → (0, ∞) with∞k=1 ψ(1k) = ∞, the set {N ∈ N: log PN (α) ≤ −ψ(log N)} has upper density 1, which answers a question of Bence Borda. On the other hand, we prove that {N ∈ N: log PN (α) ≥ ψ(log N)} has upper density at least12 , with remarkable equality if lim infk→∞ ψ(k)/(k log k) ≥ C for some sufficiently large C > 0.",

keywords = "continued fraction, Diophantine approximation, metric number theory, Sudler product",

author = "Manuel Hauke",

note = "Publisher Copyright: c 2023 American Mathematical Society.",

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doi = "10.1090/proc/16269",

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T1 - Metric density results for the value distribution of Sudler products

AU - Hauke, Manuel

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N2 - We study the value distribution of the Sudler product PN (α):=Nn=1 |2 sin(πnα)| for Lebesgue-almost every irrational α. We show that for every non-decreasing function ψ : (0, ∞) → (0, ∞) with∞k=1 ψ(1k) = ∞, the set {N ∈ N: log PN (α) ≤ −ψ(log N)} has upper density 1, which answers a question of Bence Borda. On the other hand, we prove that {N ∈ N: log PN (α) ≥ ψ(log N)} has upper density at least12 , with remarkable equality if lim infk→∞ ψ(k)/(k log k) ≥ C for some sufficiently large C > 0.

AB - We study the value distribution of the Sudler product PN (α):=Nn=1 |2 sin(πnα)| for Lebesgue-almost every irrational α. We show that for every non-decreasing function ψ : (0, ∞) → (0, ∞) with∞k=1 ψ(1k) = ∞, the set {N ∈ N: log PN (α) ≤ −ψ(log N)} has upper density 1, which answers a question of Bence Borda. On the other hand, we prove that {N ∈ N: log PN (α) ≥ ψ(log N)} has upper density at least12 , with remarkable equality if lim infk→∞ ψ(k)/(k log k) ≥ C for some sufficiently large C > 0.

KW - continued fraction

KW - Diophantine approximation

KW - metric number theory

KW - Sudler product

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Metric density results for the value distribution of Sudler products

Abstract

Keywords

ASJC Scopus subject areas

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