  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.
