  Recently, H\"ubner-Schmidt defined the tame site of a scheme. We define
$p$-adic tame Tate twists in the tame topology and prove some first properties.
We establish a framework analogous to the Beilinson-Lichtenbaum conjectures in
the tame topology for $p$-adic tame Tate twists and tame logarithmic
deRham-Witt sheaves. Both only differ from their \'etale counterpart in
cohomological degrees above the weight. These cohomology groups can be analysed
using the Gersten conjecture which, at least conjecturally, has a nice shape in
the tame topology. We prove the Gersten conjecture for tame logarithmic
deRham-Witt sheaves for curves in positive characteristic and note that the
conjecture in arbitrary dimension would follow from strict
$\mathbb{A}^1$-invariance.
