Syntomic cohomology and real topological cyclic homology

We define the motivic filtrations on real topological Hochschild homology and its companions. In particular, we prove that real topological cyclic homology admits a natural complete filtration whose graded pieces are equivariant suspensions of syntomic cohomology. As an application, we compute the equivariant slices of $p$-completed real $K$-theories of ${\mathbb{F}}_p[x]\mathrm{/}{x}^e$ and $\mathbb{Z}\mathrm{/}{p}^n$ after certain suspensions assuming a real refinement of the Dundas-McCarthy-Goodwillie theorem and an announced result of Antieau-Krause-Nikolaus.