On the Distribution of Square-Free Primitive Roots in Beatty Sequences

Let \(\alpha \) $\alpha$ be an irrational number and let \(\beta \) $\beta$ be a real number. The corresponding Beatty sequence is defined as \(\begin{aligned} \mathcal {B}_{\alpha ,\beta }=\{\lfloor \alpha n+\beta \rfloor : n\in \mathbb {N}\}, \end{aligned}\) $$\begin{array}{c}\hfill {\mathcal{B}}_{\alpha ,\beta }=\{\lfloor \alpha n+\beta \rfloor :n\in \mathbb{N}\},\end{array}$$ where \(\lfloor y\rfloor \) $\lfloor y\rfloor$ is the largest integer not exceeding y. We study the distribution of square-free primitive roots in the Beatty sequence \(\mathcal {B}_{\alpha ,\beta }\) ${\mathcal{B}}_{\alpha ,\beta }$ . As an intermediate step, bounds for character sums with square-free numbers in Beatty sequences are obtained.