Facet Connectedness Of Arithmetic Discrete Hyperplanes With Non-Zero Shift

We present a criterion for the arithmetic discrete hyper-plane P(v, mu, theta) to be facet connected when theta is the connecting thickness Omega(v, mu). We encode the shift mu in a numeration system associated with the normal vector v and we describe an incremental construction of the plane based on this encoding. We deduce a connectedness criterion and we show that when the Fully Subtractive algorithm applied to v has a periodic behaviour, the encodings of shifts mu for which the plane is connected may be recognised by a finite state automaton.