Some complete $ω$-powers of a one-counter language, for any Borel class of finite rank

Abstract: We prove that, for any natural number n $\ge$ 1, we can find a finite alphabet $\Sigma$ and a finitary language L over $\Sigma$ accepted by a one-counter automaton, such that the $\omega$-power L $\infty$ := {w 0 w 1. .. $\in$ $\Sigma$ $\omega$ | $\forall$i $\in$ $\omega$ w i $\in$ L} is $\Pi$ 0 n-complete. We prove a similar result for the class $\Sigma$ 0 n .