Lower-order refinements of greedy approximation

For two countable ordinals $\alpha $ and $\beta $, a basis of a Banach space $X$ is said to be $(\alpha , \beta )$-quasi-greedy if it is 

$\bullet$ quasi-greedy,

$\bullet$ $\mathcal {S}_\alpha $-unconditional but not $\mathcal {S}_{\alpha +1}$-unconditional, and 

$\bullet$ $\mathcal {S}_\beta $-democratic but not $\mathcal {S}_{\beta +1}$-democratic.

If $\alpha $ or $\beta $ is replaced with $\infty $, then the basis is required to be unconditonal or democratic, respectively. Previous work constructed a $(0,0)$-quasi-greedy basis, an $(\alpha , \infty )$-quasi-greedy basis, and an $(\infty , \alpha )$-quasi-greedy basis. In this paper, we construct $(\alpha , \beta )$-quasi-greedy bases for $\beta \le \alpha +1$ (except the already solved case $\alpha = \beta = 0$).