Undecidability of a weak version of MSO+U

We prove the undecidability of MSO on \(\omega\)-words extended with the second-order predicate \(U_1(X)\) which says that the distance between consecutive positions in a set \(X \subseteq \mathbb{N}\) is unbounded. This is achieved by showing that adding \(U_1\) to MSO gives a logic with the same expressive power as \(MSO+U\), a logic on \(\omega\)-words with undecidable satisfiability. As a corollary, we prove that MSO on \(\omega\)-words becomes undecidable if allowing to quantify over sets of positions that are ultimately periodic, i.e., sets \(X\) such that for some positive integer \(p\), ultimately either both or none of positions \(x\) and \(x+p\) belong to \(X\).