A classification of one dimensional almost periodic tilings arising from the projection method

For each irrational number $\alpha$, with continued fraction expansion $[0; a_1,\allowbreak a_2,a_3, \dots ]$, we classify, up to translation, the one dimensional almost periodic tilings which can be constructed by the projection method starting with a line of slope $\alpha$. The invariant is a sequence of integers in the space $X_\alpha = \{(x_i)_{i=1}^\infty \mid x_i \in \{0,1,2, \dots ,a_i\}$ and $x_{i+1} = 0$ whenever $x_i = a_i\}$ modulo the equivalence relation generated by tail equivalence and $(a_1, 0, a_3, 0, \dots ) \sim (0, a_2, 0, a_4, \dots ) \sim (a_1 -1, a_2 - 1, a_3 - 1, \dots )$. Each tile in a tiling $\textsf{T}$, of slope $\alpha$, is coded by an integer $0 \leq x \leq [\alpha]$. Using a composition operation, we produce a sequence of tilings $\textsf{T}_1 = \textsf{T}{}, \textsf{T}_2, \textsf{T}_3, \dots$. Each tile in $\textsf{T}_i$ gets absorbed into a tile in $\textsf{T}_{i+1}$. A choice of a starting tile in $\textsf{T}_1$ will thus produce a sequence in $X_\alpha$. This is the invariant.