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)⋅F(kα))k∈ℕ, {\left( {g\left( k \right) \cdot F\left( {k\alpha } \right)} \right)_{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 ℝ. All the results are stated in terms of the Diophantine properties of α, with the help of the theory of continued fractions.