Density of Oscillating Sequences in the Real Line

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.