Density of Oscillating Sequences in the Real Line
| Autor: | Tsokanos, Ioannis |
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| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | In this paper we study the density in the real line of oscillating sequences of the form $$ (g(k)\cdot F(k\alpha))_{k \in \mathbb{N}} ,$$ where $g$ is a positive increasing function and $F$ a real continuous 1-periodic function. This extends work by Berend, Boshernitzan and Kolesnik 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 $\mathbb{R}$. All the results are stated in terms of the Diophantine properties of $\alpha$, with the help of the theory of continued fractions. Comment: 17 pages |
| Databáze: | arXiv |
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