Fully Subtractive Algorithm, Tribonacci numeration and connectedness of discrete planes

We investigate connections between a well known multidimensional continued fraction algorithm, the so-called fully subtractive algorithm, the finiteness property for beta-numeration, and the connectedness of arithmetic discrete hyperplanes. A discrete hyperplane is said to be critical if its thickness is equal to the infimum of the set of thicknesses for which discrete hyperplanes with same normal vector are connected. We focus on particular planes the parameters of which belong to the cubic extension generated by the Tribonacci number, we prove connectedness in the critical case, and we exhibit an intriguing tree structure rooted at the origin.