Jones index theorem (1983) states that the set of all possible (finite) indices of subfactors is exactly $$\mathrm{Ind}=\{ 4 \cos(\pi/n)^2 \ | \ n \ge 3 \} \cup [4, \infty),$$ but if we restrict to the finite depth case, the set (say $\mathrm{Ind_{fd}}$) must become countable, because then the index is the squared norm of the principal graph (which then is a finite bipartite graph). According to this paper on page 63 (Afzaly-Morrison-Penneys, to appear in MAMS) there are exactly $8$ possible such indices in the interval $(4,5.25]$.

The set $\mathrm{Ind_{fd}}$ is multiplicative (i.e. $\alpha, \beta \in \mathrm{Ind_{fd}} \Rightarrow \alpha \beta \in \mathrm{Ind_{fd}}$) because the tensor product keeps the finite depth. Now, by this paper (Wassermann, 1998), for all $m<n$, there is a (finite depth) Jones-Wassermann subfactor of index $\frac{\sin^2(n\pi/m)}{\sin^2(\pi/m)}$, so that the set $\mathrm{Ind_{fd}}$ has an accumultation point at $\alpha n^2$ for all $n \in \mathbb{N}_{\ge 2}$ and $\alpha \in \mathrm{Ind_{fd}}$.

By this paper (Etingof-Nikshych-Ostrik, 2005), $\mathrm{Ind_{fd}}$ is contained in the set of positive cyclotomic integers (i.e. a positive elements of $\mathbb{Z}[c_n]$ with $c_n = 2\cos(\pi/n)$), which is a breakthrough in the understanding of $\mathrm{Ind_{fd}}$.

Now, I feel like that we can get even better, because by Theorem 3.2 in this paper (Bisch, 1994): $$\alpha \in \mathrm{Ind_{fd}} \setminus \mathbb{N} \Rightarrow \alpha^{-1} \mathbb{N} \cap \mathrm{Ind_{fd}} = \emptyset.$$ Example: $n(3-\sqrt{5}) \not \in \mathrm{Ind_{fd}}$ because $ 2\frac{3+\sqrt{5}}{2} \cdot n(3-\sqrt{5}) = 4n$, and $\frac{3+\sqrt{5}}{2} = 4\cos^2(\pi/5)$.

More strongly (see this comment) by extending Bisch's result to the ring of integers $R=\mathcal{O}_K$ of any cyclotomic number field $K$ (for example $R=\mathbb{Z}[c_n]$, but it can be something else): $$\alpha \in \mathrm{Ind_{fd}} \setminus R \Rightarrow \alpha^{-1} R \cap \mathrm{Ind_{fd}} = \emptyset.$$

The purpose of this post is not really to ask what exactly is $\mathrm{Ind_{fd}}$ (which seems unreachable), but new results about it, in particular inspired by above observations.