Density of Oscillating Sequences in the Real Line

TL;DR

This paper investigates the conditions under which oscillating sequences of the form (g(k)·F(kα)) are dense in the real line, extending previous work by analyzing the Diophantine properties of α and the roots of F.

Contribution

It provides necessary and sufficient conditions for the density of oscillating sequences based on the Diophantine properties of α and the roots of the periodic function F.

Findings

Sequences are dense when α satisfies certain Diophantine conditions.

F having finitely many roots is crucial for the results.

The results connect the density of sequences to continued fraction properties.

Abstract

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.