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For r = where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ Z d there exists an integer k > 0 with τ k r (a) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β-expansions and canonical number systems. After characterization… (More)

We consider the asymptotic behavior of b-additive functions f with respect to a base b of a canonical number system in the Gaussian number field. In particular, we get a normal limit law for f (P (z)) where P (z) is a polynomial with integer coefficients. Our methods are exponential sums over the Gaussian number field as well as certain results from the… (More)

Several cryptosystems rely on fast calculations of linear combinations in groups. One way to achieve this is to use joint signed binary digit expansions of small “weight.” We study two algorithms, one based on nonadjacent forms of the coefficients of the linear combination, the other based on a certain joint sparse form specifically adapted to… (More)

In this paper we study properties of the fundamental domain F of number systems, which are deened in rings of integers of number elds. First we construct addition automata for these number systems. Since F deenes a tiling of the n-dimensional vector space, we ask, which tiles of this tiling \touch" F. It turns out, that the set of these tiles can be… (More)

We consider the asymptotic behavior of the moments of the sum-of-digits function of canonical number systems in number fields. Using Delange's method we obtain the main term and smaller order terms which contain periodic fluctuations.

In the present paper we give an overview of topological properties of self-affine tiles. After reviewing some basic results on self-affine tiles and their boundary we give criteria for their local connectivity and connectivity. Furthermore, we study the connectivity of the interior of a family of tiles associated to quadratic number systems and give results… (More)

matrice d'adjacence de C interviennent. Un avantage du graphe de contact est sa structure relativement simple, ce qui rend possible sa construction immédiate pour une grande classe de substitutions. Dans cet article, nous construisons explicitement le graphe de contact pour une classe de substitutions de Pisot qui sont reliées aux β-développements par… (More)

Let T be a tile of a self-affine lattice tiling. We give an algorithm that allows to determine all neighbours of T in the tiling. This can be used to characterize the sets VL of points, where T meets L other tiles. Our algorithm generalizes an algorithm of the authors which was applicable only to a special class of self-affine lattice tilings. This new… (More)

In this paper, we study aperiodic and periodic tilings induced by the Rauzy Fractal and its subtiles associated to beta-substitutions related to the polynomial x 3 − ax 2 − bx − 1 for a ≥ b ≥ 1.