On the density of intermediate \beta-shifts of finite type

We determine the structure of the set of intermediate $\beta$-shifts of finite type. Specifically, we show that this set is dense in the parameter space $\Delta = { (\beta, \alpha) \in $\mathbb{R}$^{2} \colon \beta \in (1, 2) ; $\text{and}$ ; 0 \leq \alpha \leq 2 - \beta}$. This generalises the classical result of Parry from 1960 for greedy and (normalised) lazy $\beta$-shifts.