Undecidability of a weak version of MSO+U

We prove the undecidability of mso on ω-words extended with the second-order predicate U₁(X) which says that the distance between consecutive positions in a set X ⊆ N is unbounded. This is achieved by showing that adding U₁ to mso gives a logic with the same expressive power as mso+U, a logic on ω-words with undecidable satisfiability. As a corollary, we prove that mso on ω-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.