Generation of discrete planes

We consider the problem of generation of discrete planes using generalized substitutions. We give sufficient conditions to be sure to generate all of a discrete plane by a sequence of substitutions; these conditions, however, are not easy to check, even on simple examples. One can build approximations of discrete planes in several ways, namely as stepped surfaces (unions of faces), as sets of vertices, or as two-dimensional sequences on a three-letter alphabet; these codings are equivalent in some cases, as we recall below. Recent progresses have given a way to act on these approximations by generalized substitutions; but an open question is to know whether we can, in this way, generate all (or an arbitrarily large neighborhood of the origin) of the discrete plane. In this short communication, we give a sufficient condition to generate an arbitrarily large neighborhood of the origin; unfortunately, this condition seems, at the moment, to be difficult to check on explicit examples, even in simple cases. 1 Generation of discrete planes: definitions and known results 1.1 Discrete planes Let P(a,b,c) ⊂ R 3 be a plane with equation ax + by + cz = 0. We suppose that a, b, c > 0. We want to approximate the plane P by a stepped surface, defined as a union of faces of integral cubes. We thus introduce the discrete plane approximation P of the plane P(a,b,c) as the upper boundary of the union of all unit cubes with integral vertices that intersect this plane. This construction is inspired by the cut-and-project formalism in quasicrystals.