Growth-type invariants for ${\mathbb{Z}}^d$ subshifts of finite type and classes arithmetical of real numbers

We discuss some numerical invariants of multidimensional shifts of finite type (SFTs) which are associated with the growth rates of the number of admissible finite configurations. Extending an unpublished example of Tsirelson, we show that growth complexities of the form $\exp (n^{\alpha })$ are possible for non-integer $\alpha$’s. In terminology of Carvalho, such subshifts have entropy dimension $\alpha$. The class of possible $\alpha$’s are identified in terms of arithmetical classes of real numbers of Weihrauch and Zheng.