Facet Connectedness of Arithmetic Discrete Hyperplanes with Non-Zero Shift

摘要

We present a criterion for the arithmetic discrete hyper-plane P(υ,μ,θ) to be facet connected when θ is the connecting thickness Ω(υ,μ). We encode the shift μ 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 μ for which the plane is connected may be recognised by a finite state automaton.