Number systems over orders.

Let K be a number field of degree k and let O be an order in K . A generalized number system over O (GNS for short) is a pair ( p , D ) where p ∈ O [ x ] is monic and D ⊂ O is a complete residue system modulo p(0) containing 0. If each a ∈ O [ x ] admits a representation of the form a ≡ ∑ j = 0 ℓ - 1 d j x j ( mod p ) with ℓ ∈ N and d 0 , … , d ℓ - 1 ∈ D then the GNS ( p , D ) is said to have the finiteness property. To a given fundamental domain F of the action of Z k on R k we associate a class G F : = { ( p , D F ) : p ∈ O [ x ] } of GNS whose digit sets D F are defined in terms of F in a natural way. We are able to prove general results on the finiteness property of GNS in G F by giving an abstract version of the well-known "dominant condition" on the absolute coefficient p(0) of p. In particular, depending on mild conditions on the topology of F we characterize the finiteness property of ( p ( x ± m ) , D F ) for fixed p and large m ∈ N . Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.

On linear combinations of units with bounded coefficients and double-base digit expansions.

Let [Formula: see text] be the maximal order of a number field. Belcher showed in the 1970s that every algebraic integer in [Formula: see text] is the sum of pairwise distinct units, if the unit equation [Formula: see text] has a non-trivial solution [Formula: see text]. We generalize this result and give applications to signed double-base digit expansions.

Fractal tiles associated with shift radix systems.

Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings. In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters r these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials. We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter r of the shift radix system, these tiles provide multiple tilings and even tilings of the d-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine).