1. bookVolumen 17 (2022): Heft 1 (December 2022)
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Zeitschrift
eISSN
2309-5377
Erstveröffentlichung
30 Dec 2013
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Density of Oscillating Sequences in the Real Line

Online veröffentlicht: 31 May 2022
Volumen & Heft: Volumen 17 (2022) - Heft 1 (December 2022)
Seitenbereich: 105 - 130
Eingereicht: 21 May 2021
Akzeptiert: 27 Jan 2022
Zeitschriftendaten
License
Format
Zeitschrift
eISSN
2309-5377
Erstveröffentlichung
30 Dec 2013
Erscheinungsweise
2 Hefte pro Jahr
Sprachen
Englisch
Abstract

In this paper we study the density in the real line of oscillating sequences of the form (g(k)F(kα))k, {\left( {g\left( k \right) \cdot F\left( {k\alpha } \right)} \right)_{k \in \mathbb{N}}}, where g is a positive increasing function and F a real continuous 1-periodic function. This extends work by Berend, Boshernitzan and Kolesnik [Distribution Modulo 1 of Some Oscillating Sequences I-III] who established differential properties on the function F ensuring that the oscillating sequence is dense modulo 1.

More precisely, when F has finitely many roots in [0, 1), we provide necessary and also sufficient conditions for the oscillating sequence under consideration to be dense in ℝ. All the results are stated in terms of the Diophantine properties of α, with the help of the theory of continued fractions.

MSC 2010

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