In this paper, we study representations of real numbers in the positional numeration system with negative basis, as introduced by Ito and Sadahiro. We focus on the set Z −β of numbers whose representation uses only non-negative powers of −β, the so-called (−β)-integers. We describe the distances between consecutive elements of Z −β . In case that this set is non-trivial we associate to β an infinite word v −β over an (in general infinite) alphabet. The self-similarity of Z −β , i.e., the property −βZ −β ⊂ Z −β , allows us to find a morphism under which v −β is invariant. On the example of two cubic irrational bases β we demonstrate the difference between Rauzy fractals generated by (−β)-integers and by β-integers.
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.
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