Density of oscillating sequences in the real line

In this talk, we study the density properties in the real line of oscillating sequences of the form $( g(k) \cdot F(kα) )_{k \in \mathbb{N}}$, 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 sufficient conditions for the oscillating sequence under consideration to be dense in $\mathbb{R}$. All the related results are stated in terms of the Diophantine properties of $α$, with the help of the theory of continued fractions.