Rauzy substitution and local structure of torus tilings

For any irrational $\alpha$ we can consider tilings of the segment $[0;1]$ by the points $\{i\alpha\}$ with $0\leq i<n$. These tilings have some interesting properties, including well-known three lengths and three gaps theorems. In particular, these tilings contain segments of either two, or three different lengths. In the case of two lengths, the corresponding tilings are known as generalized Fibonacci tilings. They are closely connected with the combinatorics of words, one-dimensional quasiperiodic tilings, bounded remainder sets, first return maps for irrational circle rotations, etc.
Transferring the general three lengths and three gaps theorems to two-dimensional case, i.e. to the points $(\{i\alpha_1\},\{i\alpha_2\})$ is a well-known open problem. In the present work we consider a special case of this problem. associated with two-dimensional generalizations of Fibonacci tilings. These tilings are obtained using iterations of the geometric version of the famous Rauzy substitution. They arise in the words combinatorics in the study of generalizations of Sturmian sequences, as well as in number theory in the study of toric shifts. Considered tilings consist of rhombuses of three different types. It is proved that there are exactly 9 types of sets of rhombuses adjacent to a given rhombus. Also we obtain a method that allows explicitly determine all neighbours of the given rhombus. The results can be considered as a first step to a multidimensional generalization of the three lengths and three gaps theorems.