Lower-order refinements of greedy approximation

For two countable ordinals α and β, a basis of a Banach space X is said to be (α,β)-quasi-greedy if it is ∙ quasi-greedy, ∙ Sα-unconditional but not Sα+1-unconditional, and ∙ Sβ-democratic but not Sβ+1-democratic. If α or β is replaced with ∞, then the basis is required to be unconditonal or democratic, respectively. Previous work constructed a (0,0)-quasi-greedy basis, an (α,∞)-quasi-greedy basis, and an (∞,α)-quasi-greedy basis. In this paper, we construct (α,β)-quasi-greedy bases for β≤α+1 (except the already solved case α=β=0).