Ulrich bundles on cyclic coverings of projective spaces

We prove the existence of Ulrich bundles on cyclic coverings of $\mathbb{P}^n$ of arbitrary degree $d$. Given a relatively Ulrich bundle on a complete intersection subvariety, we construct a relatively Ulrich bundle on the ambient variety. As an application, we prove that there exists a rank $d$ Ulrich bundle on generic cyclic coverings of $\mathbb{P}^2$ of degree $d$ such that the degree of the branch divisor $d \cdot k$ is even. When $d \cdot k$ is odd, we also provide an estimation of the rank of the Ulrich bundle on generic cyclic coverings of $\mathbb{P}^2$.