Lattice Bounded Distance Equivalence for 1D Delone Sets with Finite Local Complexity

Abstract

Spectra of suitably chosen Pisot-Vijayaraghavan numbers represent non-trivial examples of self-similar Delone point sets of finite local complexity, indispensable in quasicrystal modeling. For the case of quadratic Pisot units we characterize, dependingly on digits in the corresponding numeration systems, the spectra which are bounded distance to an average lattice. Our method stems in interpretation of the spectra in the frame of the cut-and-project method. Such structures are coded by an infinite word over a finite alphabet which enables us to exploit combinatorial notions such as balancedness, substitutions and the spectrum of associated incidence matrices.

Note on BDL property of fixed points of primitive morphisms

“…This shows that the set Λ u , given by Definition 2, meets the condition on a geometric representation of an infinite word. Indeed, equations ( 1) and (2) show that the sequence (x n ) n∈Z has the property: if…”

“…In [2] authors gave following reformulation of BDL property of a geometric representation of an infinite word in terms of Parikh vectors of its prefixes. Lemma 4.…”

“…Our aim here is to describe fixed points of substitutions having non-trivial geometrical representation which is bounded distance equivalent to a lattice. A sufficient condition for this property is given by following proposition taken from [2].…”

“…In this paper we continue our study [2] of geometric representations of bi-directional infinite words…”

“…In [2] we proved two sufficient conditions for an infinite word u to have a non-trivial geometric representation with BDL property. We showed that if u is the so-called balanced word then it has a nontrivial BDL geometric representation.…”