$n$-crowns in toric tilings into bounded remander sets

The Arnoux-Ito theory of geometric substitutions allows to construct sequences of generalized exchanged tilings of the $d$-dimensional torus. These tilings consist of parallelepipeds of $d+1$ type, and the action of a certain toric shift on the tiling reduces to exchanging of the $d+1$ central parallelepipeds. Moreover, the set of vertices of all parallelepipeds of the tiling is a fragment of the orbit of zero point under considered toric shift. The considered tilings are actively used in various problems of number theory, combinatorics, and the theory of dynamical systems.
In this paper, we study the local structure of toric tilings obtained using geometric substitutions. The $n$-corona of the parallelepiped is a set of all parallelepipeds located at a distance of not greater than $n$ from a given parallelepiped in the natural metric of the tiling. The problem is to describe all possible types of $n$-coronas.
With each parallelepiped in the tiling we can naturally assigned a number — its number in the orbit of the corresponding central parallelepiped with respect to the toric shift. It is proved that the set of all parallelepipeds numbers splits into a finite number of half-intervals defining possible types of $n$-coronas. Moreover, it is proved that the boundaries of the corresponding half-open intervals are determined by the numbers of the parallelepipeds in the $n$-corona of the set of $d+1$ central parallelepiped.
It is shown that this result can be considered as some multi-dimensional generalization of the famous three lengths theorem. Earlier, a similar description was obtained for 1-coronas of the toric tilings obtained using one specific geometric substitution: the Rauzy substitution. In addition, similar results were previously obtained for some quasiperiodic plane tilings.
In conclusion, some directions for further research are formulated.