A slice refinement of B\"okstedt periodicity

Let $R$ be a perfectoid ring. Hesselholt and Bhatt-Morrow-Scholze have identified the Postnikov filtration on $\mathrm{THH}(R;\mathbb Z_p)$: it is concentrated in even degrees, generated by powers of the B\"okstedt generator $\sigma$, generalizing classical B\"okstedt periodicity for $R=\mathbb F_p$. We study an equivariant generalization of the Postnikov filtration, the *regular slice filtration*, on $\mathrm{THH}(R;\mathbb Z_p)$. The slice filtration is again concentrated in even degrees, generated by $RO(\mathbb T)$-graded classes which can loosely be thought of as the *norms* of $\sigma$. The slices are expressible as $RO(\mathbb T)$-graded suspensions of Mackey functors obtained from the Witt Mackey functor. We obtain a sort of filtration by $q$-factorials. A key ingredient, which may be of independent interest, is a close connection between the Hill-Yarnall characterization of the slice filtration and Ansch\"utz-le Bras' $q$-deformation of Legendre's formula.