Critical connectedness of thin arithmetical discrete planes

An arithmetical discrete plane is said to have critical connecting thickness if its thickness is equal to the infimum of the set of values that preserve its $2$-connectedness. This infimum thickness can be computed thanks to the fully subtractive algorithm. This multidimensional continued fraction algorithm consists, in its linear form, in subtracting the smallest entry to the other ones. We provide a characterization of the discrete planes with critical thickness that have zero intercept and that are $2$-connected. Our tools rely on the notion of dual substitution which is a geometric version of the usual notion of substitution acting on words. We associate with the fully subtractive algorithm a set of substitutions whose incidence matrix is provided by the matrices of the algorithm, and prove that their geometric counterparts generate arithmetic discrete planes.Comment: 18 pages, v2 includes several corrections and is a long version of the DGCI extended abstrac

On the Connectedness of Rational Arithmetic Discrete Hyperplanes

Abstract. While connected arithmetic discrete lines are entirely characterized by their arithmetic thickness, only partial results exist for arithmetic discrete hyperplanes in any dimension. In the present paper, we focus on 0-connected rational arithmetic discrete planes in Z 3. Thanks to an arithmetic reduction on a given integer vector n, we provide an algorithm which computes the thickness of the thinnest 0-connected arithmetic plane with normal vector n.