Rokhlin dimension for compact quantum group actions

We show that, for a given compact or discrete quantum group $G$, the class of actions of $G$ on C*-algebras is first-order axiomatizable in the logic for metric structures. As an application, we extend the notion of Rokhlin property for $G$-C*-algebra, introduced by Barlak, Szab'{o}, and Voigt in the case when $G$ is second countable and coexact, to an arbitrary compact quantum group $G$. All the the preservations and rigidity results for Rokhlin actions of second countable coexact compact quantum groups obtained by Barlak, Szab'{o}, and Voigt are shown to hold in this general context. As a further application, we extend the notion of equivariant order zero dimension for equivariant -homomorphisms, introduced in the classical setting by the first and third authors, to actions of compact quantum groups. This allows us to define the Rokhlin dimension of an action of a compact quantum group on a C-algebra, recovering the Rokhlin property as Rokhlin dimension zero. We conclude by establishing a preservation result for finite nuclear dimension and finite decomposition rank when passing to fixed point algebras and crossed products by compact quantum group actions with finite Rokhlin dimension.