In this paper we study the density in the real line of oscillating sequences
of the form

\[
\bigl(g(k)\cdot F(k\alpha)\bigr)_{k\in\mathbb{N}},
\]

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 \( \mathbb{R} \). All the results are stated in terms
of the Diophantine properties of \( \alpha \), with the help of the theory of
continued fractions.