Number representation and finite automata

Dynamical directions in numeration

“…The question of renormalisation (or change of alphabet) is motivated by performing arithmetic operations. In [148,149], Frougny shows among other things that renormalisation is computable by a finite transducer in the case of a G-scale given by a linear recurrence sequence (G-scales are introduced and discussed in Section 5). These survey papers develop the theory of β-representation from the point of view of automata theory.…”

“…An extensively studied question is to find the characterisation of these β (see Section 4). More generally, for detailed surveys on the β-numeration, see for instance [69,74,242,148,304]. …”

Diffraction spectra of weighted Delone sets on beta-lattices with beta a quadratic unitary Pisot number

“…We refer to [15] [16], Chapter 7 in [26], [33] [35] for the numeration in Pisot base, and to [20] [24] [28] [29] [30] [32] [41] [42] for an introduction to mathematical quasicrystals (to Delone sets, Meyer sets ...).…”

“…The aim of this paper is first to recall the part of this program specifically devoted to the notions of sets of beta-integers ( Z β ), when β is a quadratic Pisot-Vijayaraghavan number, and associated beta-lattices (or β-lattices), and next to present some original results concerning the Fourier properties and diffraction spectra of Delone sets on beta-integers and on beta-lattices, and more generally of weighted Dirac combs supported by beta-lattices. This approach makes prominent the role played by nonclassical numerations [16] [26], associated with substitutive dynamical systems [34], and amounts to replace the traditional set of rational integers Z by Z β . This allows computations over this new set of numbers, where lattices are replaced by beta-lattices.…”

Combinatorial and probabilistic properties of systems of numeration

“…, a d−1 . Numeration based on linear recurrent scales have been studied in [14][15][16][17][18][19][20] with respect to the representability of the addition-by-one map by a finite automaton. In the context of linear recursive base sequences the case when the characteristic polynomial…”