BooksirmaCollected Volumepp. 79–131

Nested quasicrystalline discretisations of the line

  • Jean-Pierre Gazeau

    Université Denis Diderot - Paris 7, France
  • Zuzana Masáková

    Czech Technical University, Praha, Czech Republic
  • Edita Pelantová

    Czech Technical University, Praha, Czech Republic
Nested quasicrystalline discretisations of the line cover
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Abstract

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, combinatorial 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.