Nested Quasicrystalline Discretisations of the Line

One-dimensional cut-and-project point sets obtained from the square lattice in the plane are considered from a unifying point of view and in the perspective of aperiodic wavelet constructions. We successively examine their geometrical aspects, combinato-rial properties from the point of view of the theory of languages, and self-similarity with algebraic scaling factor θ. We explain the relation of the cut-and-project sets to non-standard numeration systems based on θ. We finally examine the substitutivity, a weakened version of substitution invariance, which provides us with an algorithm for symbolic generation of cut-and-project sequences.