$p$-adic tame Tate twists

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.