Classification of 2-quasi-invariant subsets

Let G be a group acting on a set Ω. A subset (finite or infinite) A ⊆ Ω is called k-quasi-invariant, where k is a non-negative integer, if |A⁹ \ A| ≤ k for every g ∈ G. In previous work of the authors a bound was obtained, in terms of k, on the size of the symmetric difference between a k-quasi-invariant subset and the G-invariant subset of Ω closest to it. However, apart from the cases k = 0, 1, this bound gave little information about the structure of a k-quasi-invariant subset. In this paper a classification of 2-quasi-invariant subsets is given. Besides the generic examples (subsets of Ω which have a symmetric difference of size at most 2 with some G-invariant subset) there are basically five explicitly determined possibilities.