Actions of tame abelian product groups

A Polish group $G$ is tame if for any continuous action of $G$, the corresponding orbit equivalence relation is Borel. When $G = \prod_n \Gamma_n$ for countable abelian $\Gamma_n$, Solecki (1995) gave a characterization for when $G$ is tame. Ding and Gao (2017) showed that for such $G$, the orbit equivalence relation must in fact be potentially $\mathbf{\Pi}^0_6$, while conjecturing that the optimal bound could be $\mathbf{\Pi}^0_3$. We show that the optimal bound is $D(\mathbf{\Pi}^0_5)$ by constructing an action of such a group $G$ which is not potentially $\mathbf{\Pi}^0_5$, and show how to modify the analysis of Ding and Gao to get this slightly better upper bound. It follows, using the results of Hjorth, Kechris, and Louvaeu (1998), that this is the optimal bound for the potential complexity of actions of tame abelian product groups. Our lower-bound analysis involves forcing over models of set theory where choice fails for sequences of finite sets.