Complemented sets, difference sets, and weakly wandering sequences

Abstract

We consider the descriptive complexity of several sets of sequences of natural numbers, and show that the following are all complete analytic sets: the set of complemented sequences, the set of sequences containing an infinite difference set, the set of sequences which are weakly wandering sequences for some transformation, and several variants of these. We then use the same techniques to produce weakly wandering sequences with special properties, such as a sequence which is exhaustive weakly wandering for some transformation but which is not weakly wandering for any ergodic transformation. In this paper we consider descriptive aspects of weakly wandering sequences. These sequences are isomorphism invariants for measure-preserving transformations or Borel automorphisms introduced by Hajian and Kakutani in [6]. We first consider how difficult it is to determine whether some sequence can be a weakly wandering sequence for some transformation, a