1. bookVolumen 17 (2022): Edición 1 (December 2022)
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eISSN
2309-5377
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30 Dec 2013
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2 veces al año
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Density of Oscillating Sequences in the Real Line

Publicado en línea: 31 May 2022
Volumen & Edición: Volumen 17 (2022) - Edición 1 (December 2022)
Páginas: 105 - 130
Recibido: 21 May 2021
Aceptado: 27 Jan 2022
Detalles de la revista
License
Formato
Revista
eISSN
2309-5377
Primera edición
30 Dec 2013
Calendario de la edición
2 veces al año
Idiomas
Inglés
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.

Keywords

MSC 2010

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