Number systems over orders

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {K}}$$\end{document}K be a number field of degree k and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}$$\end{document}O be an order in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {K}}$$\end{document}K. A generalized number system over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}$$\end{document}O (GNS for short) is a pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p,{\mathcal {D}})$$\end{document}(p,D) where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in {\mathcal {O}}[x]$$\end{document}p∈O[x] is monic and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}\subset {\mathcal {O}}$$\end{document}D⊂O is a complete residue system modulo p(0) containing 0. If each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \in {\mathcal {O}}[x]$$\end{document}a∈O[x] admits a representation of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \equiv \sum _{j =0}^{\ell -1} d_j x^j \pmod {p}$$\end{document}a≡∑j=0ℓ-1djxj(modp) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \in {\mathbb {N}}$$\end{document}ℓ∈N and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_0,\ldots , d_{\ell -1}\in {\mathcal {D}}$$\end{document}d0,…,dℓ-1∈D then the GNS \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p,{\mathcal {D}})$$\end{document}(p,D) is said to have the finiteness property. To a given fundamental domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}$$\end{document}F of the action of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}^k$$\end{document}Zk on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^k$$\end{document}Rk we associate a class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {G}}_{\mathcal {F}} := \{ (p, D_{\mathcal {F}}) \;:\; p \in {\mathcal {O}}[x] \}$$\end{document}GF:={(p,DF):p∈O[x]} of GNS whose digit sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{\mathcal {F}}$$\end{document}DF are defined in terms of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}$$\end{document}F in a natural way. We are able to prove general results on the finiteness property of GNS in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {G}}_{\mathcal {F}}$$\end{document}GF 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}$$\end{document}F we characterize the finiteness property of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p(x\pm m), D_{\mathcal {F}})$$\end{document}(p(x±m),DF) for fixed p and large \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\in {\mathbb {N}}$$\end{document}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.


Introduction
In the present paper we introduce a general notion of number system defined over orders of number fields. This generalizes the well-known number systems and canonical number systems in number fields to relative extensions and allows for the definition of "classes" of digit sets by the use of lattice tilings. It fits in the general framework of digit systems over commutative rings defined by Scheicher et al. [32].
Before the beginning of the 1990s canonical number systems have been defined as number systems that allow to represent elements of orders (and, in particular, rings of integers) in number fields. After the pioneering work of Knuth [23] and Penney [28], special classes of canonical number systems have been studied by Kátai and Szabó [22], Kátai and Kovács [20,21], and Gilbert [15], while elements of a general theory are due to Kovács [24] as well as Kovács and Pethő [25,26]. In 1991 Pethő [29] gave a more general definition of canonical number systems that reads as follows. Let p ∈ Z[x] be a monic polynomial and let D be a complete residue system of Z modulo p(0) containing 0. The pair ( p, D) was called a number system if each a ∈ Z[x] allows a representation of the form If such a representation exists it is unique if we forbid "leading zeros", i.e., if we demand d −1 = 0 for a ≡ 0 (mod p) and take the empty expansion for a ≡ 0 (mod p) (note that the fact that 0 ∈ D is important for the unicity of the representation). It can be determined algorithmically by the so-called "backward division mapping" (see e.g. [1,Section 3] or [32,Lemma 2.5]). Choosing the digit set D = {0, 1, . . . , | p(0)| − 1}, the pair ( p, D) was called a canonical number system, CNS for short. An overview about the early theory of number systems can be found for instance in Akiyama et al. [1] and Brunotte et al. [10]. Let p ∈ Z[x] and let D be a complete residue system modulo p(0). With the development of the theory of radix representations it became necessary to notationally distinguish an arbitrary pair ( p, D) from a particular pair ( p, D) for which each a ∈ Z[x] admits a representation of the form (1.1). Nowadays in the literature an arbitrary pair ( p, D) is called number system (or canonical number system if D = {0, 1, . . . , | p(0)| − 1}), while the fact that each a ∈ Z[x] admits a representation of the form (1.1) is distinguished by the suffix with finiteness property. Although there exist many partial results on the characterization of number systems with finiteness property with special emphasis on canonical number systems (see for instance [2,3,7,9,12,24,33,35]), a complete description of this property seems to be out of reach (although there are fairly complete results for finite field analogs which can be found e.g. in [5,13,27]).
If ( p, D) is a number system and a ∈ Z[x] admits a representation of the form (1.1), we call the length of the representation of a in this number system (for a ≡ 0 (mod p) this length is zero by definition).
In the present paper we generalize the CNS concept in two ways. Firstly, instead of looking at polynomials in Z[x] we consider polynomials with coefficients in some order O of a given number field of degree k, and secondly, we consider the sets of digits in a more general but uniform way (see Definition 2.1). Indeed, for each fundamental domain F of the action of Z k on R k we define a class of number systems ( p, D F ) where F associates a digit set D F with each polynomial p ∈ O[x] in a natural way. Thus for each fundamental domain F we can define a class G F := {( p, D F ) : p ∈ O[x]} of number systems whose properties will be studied.
Our main objective will be the investigation of the finiteness property (see Definition 2.7) for these number systems. For a given pair ( p, D) this property can be checked algorithmically (cf. Theorem 2.9). This makes it possible to prove a strong bound for the length of the representations (given in Theorem 2.10), provided it exists.
The "dominant condition", a condition for the finiteness property of ( p, D) that involves the largeness of the absolute coefficient of p, has been studied for canonical number systems in several versions for instance in Kovács [24, Section 3], Akiyama and Pethő [2, Theorem 2], Scheicher and Thuswaldner [33,Theorem 5.8], and Pethő and Varga [31,Lemma 7.3]. The main difficulty of the generalization of the dominant condition is due to the fact that in O we do not have a natural ordering, hence, we cannot adapt the methods that were used in the case of integer polynomials. However, by exploiting tiling properties of the fundamental domain F we are able to overcome this difficulty, and provide a general criterion for the finiteness property (see Theorem 3.1) that is in the spirit of the dominant condition and can be used in the proofs of our main results. In particular, using this criterion, depending on natural properties of F we are able to show that ( p(x + m), D F ) enjoys a finiteness property for each given p provided that m (or |m|) is large enough. This forms a generalization of an analogous result of Kovács [24] to this general setting (see Theorem 4.1 and its consequences). In Theorem 5.2 we give a converse of this result by showing that ( p(x − m), D F ) doesn't enjoy the finiteness property for large m if F has certain properties. If for any root α of p. Thus in this case the finiteness property of ( p, D) is easily seen to be equivalent to the fact that each γ ∈ Z[α] admits a unique expansion of the form with analogous conditions on d 0 , . . . , d −1 ∈ D as in (1.1). In this case we sometimes write (α, D) instead of ( p, D), see Sect. 6 (note that | p(0)| = |N Q(α)/Q (α)|). This relates number systems to the problem of power integral bases of orders. Recall that the order O has a power integral basis, if there exists α ∈ O such that each γ ∈ O can be written uniquely in the form γ = g 0 + g 1 α + · · · + g k−1 α k−1 with g 0 , . . . , g k−1 ∈ Z. In this case O is called monogenic. The definitions of number system with finiteness property (1.2) and power integral bases seem similar and indeed there is a strong relation between them. Kovács [25] proved that if 1, α, . . . , α k−1 is a power integral basis then, up to finitely many possible exceptions, (α − m, N 0 (α − m)), m ∈ Z is a CNS with finiteness property if and only if m > M(α), where M(α) denotes a constant. A good overview over this circle of ideas is provided in the book of Evertse and Győry [14]. Using Theorem 4.1 we generalize the results of Kovács [24] and of Kovács and Pethő [25] to number systems over orders in algebraic number fields, see especially Corollary 4.3. Our result is not only more general as the earlier ones, but sheds fresh light to the classical case of number systems over Z too. It turns out (see Theorem 6.2) that under general conditions in orders of algebraic number fields the power integral bases and the bases of number systems with finiteness condition up to finitely many, effectively computable exceptions coincide. Choosing for example the symmetric digit set, the conditions of Theorem 6.2 are satisfied and, hence, power integral bases and number systems coincide up to finitely many exceptions. This means that CNS are quite exceptional among number systems.

Number systems over orders of number fields
In this section we define number systems over orders and study some of their basic properties. This new notion of number system generalizes the well-known canonical number systems defined by Pethő [29] that we mentioned in the introduction. Before we give the exact definition, we introduce some notation.
Let K be a number field of degree k. The field K has k isomorphisms to C, whose images are called its (algebraic) conjugates and are denoted by K (1) , . . . , K (k) . We denote by α ( j) the image of α ∈ K in K ( j) , j = 1, . . . , k. Let O be an order in K, i.e., a ring which is a Z-module of full rank in K.  Remark 2.2 Note that GNS fit in the more general framework of digit systems over a commutative ring defined in Scheicher et al. [32]. Indeed, [32, Example 6.6] provides an example of a digit system over a commutative ring that corresponds to the case O = Z[i] of our definition and uses rational integers as digits. Our more specialized setting allows us to prove results that are not valid for arbitrary commutative rings. Let be a Z-basis of O and arrange this basis in a vector Let F be a bounded fundamental domain for the action of Z k on R k , i.e., a set that satisfies R k = F + Z k without overlaps, and assume that 0 ∈ F. Such a fundamental domain defines a set of digits in a natural way. Indeed, for ϑ ∈ O define . . , f k ) ∈ F . Note that D F ,ϑ depends on the vector ω, i.e., on the basis (2.1).

Lemma 2.3
For ϑ ∈ O the set D F ,ϑ is a complete residue system modulo ϑ.
Proof Let β ∈ O. Then, because β ϑ ∈ K, and the entries of ω form a Q-basis of K, Let now τ 1 , τ 2 ∈ D F ,ϑ be given in a way that τ 1 ≡ τ 2 (mod ϑ). Then there is r ∈ F such that τ ϑ = r · ω for ∈ {1, 2}. As Proof Let ι : K → R k be the embedding defined by r · ω → r. Then there is a unique matrix P satisfying ι( It is easily verified that this is a fundamental domain satisfiying D = D F , p(0) .
If the polynomial p is clear from the context we will use the abbreviation We call a fundamental domain F satisfying the claim of Lemma 2.4 a fundamental domain associated with ( p, D). In view of this lemma we may assume in the sequel that a number system ( p, D) has an associated fundamental domain F. On the other hand, a fixed fundamental domain F defines a whole class of GNS, namely, Example 2. 5 We consider some special choices of F corresponding to families G F studied in the literature.
Classical CNS Canonical number systems as defined in the introduction form a special case of GNS. Indeed let K = Q and O = Z. Then k = 1, since Q has degree 1 over Q. Now we choose F = [0, 1) which obviously is a fundamental domain of Z acting on R. We look at the class which is the digit set of a canonical number system, we see that the class G F coincides with the set of canonical number systems in this case. If, however, ϑ ∈ Z with ϑ ≤ −2 then Symmetric CNS Symmetric CNS are defined in the same way as CNS, apart from the digit set. Indeed, These number systems were studied for instance by Akiyama and Scheicher [4, Section 2] and Brunotte [8] (see also Kátai [19,Theorem 7] for a slightly shifted version and Scheicher et al. [32,Definition 5.5] for a more general setting). They are easily seen to be equal to the class 11} be an Euclidean quadratic field with ring of integers (i.e., maximal order) O and set (this set looks a bit like a sail) one immediately checks that in Pethő and Varga [31] the class of GNS even yields a class of GNS for any imaginary quadratic number field. The square As a last example we mention the number systems over Z[i] studied by Jacob and Reveilles [18] and Brunotte et al. [11]. They correspond to the class A fundamental domain F induces by definition a tiling of R k by Z k -translates which in turn induces the following neighbor relation on Z k . We call z ∈ Z k a neighbor of be the set of neighbors of 0. We need the following easy result.

Lemma 2.6 The set of neighbors N of F contains a basis of the lattice Z k .
Proof Assume that this is wrong and let ∼ be the transitive hull of the neighbor relation on Z k . It is easy to see that this is an equivalence relation. By assumption there is z ∈ Z k such that 0 z and, hence, there are at least two equivalence classes of ∼. Let C be one of them. Then C and Z k \C are contained in pairwise disjoint unions of equivalence classes. Since F is bounded and the union z∈Z k (F + z) is locally finite this implies that the nonempty sets A = z∈C (F +z) and B = z∈Z k \C (F +z) satisfy A ∩ B = ∅ and, by the tiling property, A ∪ B = R k . This is absurd because it would imply that R k is disconnected.
If d −1 = 0 or = 0 (which results in the empty sum) then is called the length of the representation of a. It will be denoted by L(a). A "good" number system admits finite digit representations of all elements. We give a precise definition for GNS having this property.

Definition 2.7 (Finiteness property) Let ( p, D) be a GNS and set
As in the case O = Z with canonical digit set (see [29, Theorem 6.1(i)] and [25,Theorem 3]), also in our general setting the finiteness property of ( p, D) implies expansiveness of the basis p in the sense stated in the next result. (Note that its proof is also reminiscent of the proof of Vince [34,Proposition 4]; this is a related result in the context of self-replicating tilings.) Inserting α in the last congruence we get m = a ( j) (α). As D ( j) is a finite set and |α| < 1 the set of the numbers |m| = |a ( j) (α)| is bounded, which is a contradiction to m being an arbitrary rational integer. Since j was arbitrary, |α| ≥ 1 has to hold for all roots of Assume now that |α| = 1 holds for a root α of p ( j) (x) for some j ∈ {1, . . . , k}. The element α is an algebraic integer, and a root of the polynomial , and equivalently ). Let t denote the degree of a(x). Choosing h so that hs > t we obtain which contradicts the uniqueness of a ( j) (x).
Proof The necessity assertion is an immediate consequence of Proposition 2.8, hence, we have to prove only the sufficiency assertion.
be the backward division mapping, which is defined as In the remaining part of the proof, for the sake of simplicity we assume that p is Taking absolute values, choosing h large enough, and using the fact that |α i | > 1 we obtain h j x j , i = 1, . . . , k. Considering (2.7) as a system of inequalities in the unknowns d Indeed, this is true because (2.7) says that all the Galois conjugates of the element 11 ] are bounded by the explicit bounds given in (2.7). This is true only for finitely many elements of the order O[α 11 ] in the field K(α 11 ) (which has degree k deg p over Q), and these elements can be explicitly computed. Choosing Theorem 2.9 implies that the GNS property is algorithmically decidable. One has to apply the backward division mapping defined above to all polynomials satisfying deg a < deg p and H (a) ≤ C iteratively. During the iteration process one always works with polynomials satisfying these inequalities. More on algorithms for checking the finiteness property of GNS can be found in a more general context in Scheicher et al. [32,Section 6].
The proof of Theorem 2.9 makes it possible to prove a precise bound for the length of a representation in a GNS ( p, D) with finiteness property. This is in complete agreement with an analogous result of Kovács and Pethő [26] for the case O = Z.
There exist only finitely many With a little more effort one could replace irreducibility of p by separability of p in the statement of Theorem 2.10.

A general criterion for the finiteness property
There exist some easy-to-state sufficient conditions for the finiteness property of a CNS ( p, D) in the case O = Z, see e.g. Kovács Lemma 7.3]. In each of these results | p(0)| dominates over the other coefficients of p. In general, O does not have a natural ordering. However, inclusion properties of some sets can be used to express dominance of coefficients in O. This is the message of the Theorem 3.1, which will be proved in this section. Before we state it, we introduce some notation.
For p(x) = x n + p n−1 x n−1 + · · · + p 0 ∈ O[x] let ( p, D) be a GNS and let F be an associated fundamental domain. Let the basis ω 1 = 1, ω 2 , . . . , ω k be given as in (2.1), set ω = (ω 1 , . . . , ω k ) as in (2.2), and recall the definition of the set N of neighbors of 0 in (2.4). Set (letting p n = 1) = N · ω and Z = n j=1 δ j p j : δ j ∈ , (3.1) and note that, since F is bounded, these sets are finite. ( p, D) be a GNS. Let F be an associated fundamental domain and define and Z as in (3.1). Assume that the following conditions hold (setting p n = 1):

Then ( p, D) has the finiteness property.
We note that in the statement of Theorem 3.1 the set from (3.1) can be replaced by an arbitrary set that contains 0 and generates O as a semigroup and the result still remains true by the same proof. Since we only need Theorem 3.1 for our particular choice of we stated the theorem for this particular case.
To prove Theorem 3.1 we need the following auxiliary result.

Lemma 3.2 The GNS ( p, D) has the finiteness property if and only if for each a ∈ R( p, D) and each α ∈ we have a + α ∈ R( p, D).
Proof The necessity of the condition is obvious, so we are left with proving its sufficiency. Assume that for each a ∈ R( p, D) and each α ∈ we have a + α ∈ R( p, D). By Lemma 2.6, the set generates O as a semigroup. Thus in order to prove the finiteness property it is sufficient to show that a ∈ R( p, D) implies that a + αx m ∈ R( p, D) for each α ∈ and each m ≥ 0. (3. 2) The case m = 0 is true by assumption. Now choose m ≥ 1. Let a ∈ R( p, D). To conclude the proof we have to show that a + αx m ∈ R( p, D) holds for each α ∈ . We may write a(x) ≡ −1 D). Since α ∈ , and (3.2) holds for m = 0 we haveã(x) + α ∈ R( p, D) as well and, hence, (3.3) implies that a(x) + αx m ∈ R( p, D).
After this preparation we turn to the proof of Theorem 3.1.

Proof of Theorem 3.1 Our goal is to apply Lemma 3.2. To this end let a ∈ R( p, D)
and α ∈ be given. We have to show that a(x) + α ∈ R( p, D).
Since a ∈ R( p, D) we may write a(x) ≡ −1 j=0 d j x j (mod p) with d 0 , . . . , d −1 ∈ D. For convenience, in what follows we set d j = 0 for j ≥ , p n = 1, and p j = 0 for j > n. Then a(x) ≡ ∞ j=0 d j x j (mod p). Since α + d 0 ∈ Z + D (note that ⊂ Z ), condition (i) implies that there is δ 0 ∈ and b 0 ∈ D such that α We want to prove that for each t ≥ 0 the sum a(x) + α can be written in the form with b j ∈ D and δ j ∈ for 0 ≤ j ≤ t. Indeed, we prove this by induction. Since this is true for t = 0 by (3.4) assume that it is true for some given value t ≥ 0. The coefficient of x t+1 in (3.5) is d t+1 + s with s = δ 0 p t+1 + δ 1 p t + · · · + δ t p 1 .
As p j = 0 for j > n the sum s has at most n nonzero summands each of which is of the form δ j p t+1− j with δ j ∈ and t − n + 1 ≤ j ≤ t. Thus s ∈ Z and, hence, d t+1 + s ∈ D + Z . Now by condition (i) there exists b t+1 ∈ D and δ t+1 ∈ such that Thus, adding δ t+1 p(x)x t+1 to (3.5) we obtain a similar expression for a(x) + α with t replaced by t + 1. Thus, by induction, (3.5) holds for all t ≥ 0. Note that the sum in (3.5) is finite since p j = 0 for j > n. Assume now that t ≥ − 1 in (3.5). Then for j ≥ t + 1 we have d j = 0 and, hence, the coefficient of x j has the form δ 0 p j + δ 1 p j−1 + · · · + δ t p j−t ∈ Z . By (ii) this implies that δ 0 p j + δ 1 p j−1 + · · · + δ t p j−t ∈ D ∪ (D − p 0 ). This entails that δ j ∈ {0, 1} for j ≥ t + 1. Hence, if t ≥ − 1 + n for each of the nonzero summands of δ 0 p j +δ 1 p j−1 +· · ·+δ t p j−t the coefficient δ i equals 1 and thus the sum belongs to D by (iii). Consequently, in the representation (3.5) for t ≥ − 1 + n all the coefficients belong to D and, since this sum is finite, a(x) + α ∈ R( p, D). Thus the condition of Lemma 3.2 is satisfied and we may apply the lemma to conclude that ( p, D) is an GNS with finiteness property. This proves the theorem.

The finiteness property for large constant terms
One of the main results of this paper is a generalization of a result of B. Kovács [24, Section 3] that will be stated and proved in the present section. We begin with some notation. We denote by e 1 = (1, 0, . . . , 0) ∈ R k the first canonical basis vector of R k . Let M ⊂ R k . For ε > 0 we set (M) ε := {x ∈ R k : ||x − y|| ∞ < ε for some y ∈ M} for the ε-neighborhood of a set M. Moreover, int + is the interior taken w.r.t. the subspace topology on {(r 1 , . . . , r k ) ∈ R k : r 1 ≥ 0}. The symbol int − is defined by replacing r 1 ≥ 0 with r 1 ≤ 0.

Remark 4.2 Note that this implies that for each bounded fundamental domain
the family G F of GNS contains infinitely many GNS with finiteness property.
Proof Our goal is to apply Theorem 3.1.
Let j 0 be an index with r = |r j 0 |. For this index we have Using (4.5) this implies that r = |r j 0 | tends to zero for η → 0. Thus for η small enough we have r < ε with ε as in (4.1) and, hence, ||r|| ∞ = ||(r 1 , . . . , r k )|| ∞ < ε. By the choice of ε this implies that hold for η small enough. Multiplying both relations in (4.7) by p 0 (α) · ω this yields by (4.6) and the definition of in (3.1) that hold for η small enough. Intersecting the relations in (4.8) with O, and using the definition of D F , p 0 (α) in (2.3) this implies that hold for η small enough. Since ζ ∈ Z was arbitrary we have shown that there is and This implies conditions (i) and (ii) of Theorem 3.1.
Proof Again we want to apply Theorem 3.1. Choose ε 1 > 0 in a way that the ε 1 -ball around 0 w.r.t. || · || ∞ is contained in int(F). Since the union F + Z k is a locally finite union of bounded sets, the definition of the neighbor set N implies that there exists ε 2 > 0 such that (F) ε 2 ∩ (F + z) = ∅ for each z / ∈ N . Let now ε = min{ε 1 , ε 2 }. Define Let now ζ ∈ Z = Z (α) be given. In the same way as we showed (4.9) and (4.10) in the proof of Theorem 4.1 we can show, using ε as defined above, that hold for small η. Thus, since ζ ∈ Z was arbitrary and Z ⊂ Z , there is η 1 > 0 with In this case, F needs to be translated appropriately in order to make our results applicable.

Moreover, because
We will see in the next section that ( p(x − m), F) doesn't have the finiteness property for large m under the conditions of Theorem 4.1.
Before this we deal with the following conjecture of Akiyama, see Brunotte [6]: let p ∈ Z[x] be a CNS polynomial. Then there exists M such that p(x) + m is a CNS polynomial for all m ≥ M. Theorem 4.1 implies results concerning this conjecture even for polynomials over orders.
Proof Repeat the proof of Corollary 4.3 with p(x) ± α instead of p(x ± α).

GNS without finiteness property
The main result of this section complements the results of Sect. 4. We start with a partial generalization of [25,Theorem 3] to polynomials with coefficients of O that will be needed in its proof. ( p, D) be a GNS. If there exist h ∈ N, d 0 , d 1 , . . . , d h−1 ∈ D not all equal to 0 and q 1 ,

Lemma 5.1 Let
holds for all ∈ N. Since D is a complete residue system modulo p(0) this implies that a possible finite digit representation must satisfy L ≥ h for all ∈ N. Thus L cannot be finite, a contradiction. This implies that ( p, D) does not have the finiteness property.
Our main result in this section is the following theorem.
is not a GNS with finiteness property whenever m is large enough.
It remains to prove the claim. Let p(x) = x n + p n−1 x n−1 + · · · + p 0 . Then This is a system of linear equations in the unknowns r m j , j = 1, . . . , k with coefficient matrix (ω (i) j ) i, j=1,...,k . As ω 1 , . . . , ω k is a basis of O the determinant of (ω (i) j ) is not zero. Moreover, as ω 1 = 1 the first column of (ω (i) j ) is 1, the vector which consists only of ones.
We estimate the solutions of (5.4) by using Cramer's rule. If j > 1 then to get the matrix of the numerator we have to replace the j-th column of (ω (i) j ) by the vector This yields that 1 − n 2m < r m1 < 1 (5.6) holds for m large. Thus, since 0 ∈ int − (F − e 1 ) by assumption, (5.5), and (5.6) impliy that (r m1 , . . . , r mk ) ∈ F for large m and, hence, m (0) ∈ D F , p(−m−1) for large m, and the claim is proved.

GNS in number fields
As mentioned in the introduction, the theory of generalized number systems started with investigations in the ring of integers of algebraic number fields. For an overview on related results we refer to Evertse and Győry [14] and Brunotte, Huszti, and Pethő [10]. To clarify the connection of our investigations with the earlier ones we need some definitions. Let L be a number field of degree l, and denote its ring of integers by O L . Let α ∈ O L and let N be a , the finiteness property of (α, N ) coincides with the finiteness property of ( p, N ) according to the introduction.) Kovács [24] proved that there exists a canonical number system with finiteness property in O L if and only if O L admits a power integral bases. Later Kovács and Pethő [25] proved the stronger result. From Corollary 4.3 we derive that for number systems the relation is usually stronger, the theorem of Kovács and Pethő describes a kind of "boundary case" viz. a case where 0 ∈ ∂F. Theorem 6.2 Let L be a number field of degree l and let O be an order in L. Let F be a bounded fundamental domain for the action of Z on R. If 0 ∈ int(F) then all but finitely many generators of power integral bases of O form a basis for a number system with finiteness property. Moreover, the exceptions are effectively computable.
Proof By Győry [16] in O there exist up to translations by integers only finitely many generators of power integral bases, and they are effectively computable. Denote these finitely many generators by α 1 , . . . , α t and denote the minimal polynomial of α j by p j (x), j = 1, . . . , t. Fix 1 ≤ j ≤ t. Note that p j (x) is monic and has rational integer coefficients. By Corollary 4.3 (see especially Remark 4.4) there exists M j ∈ Z, such that ( p j (x ± m), F) is a GNS with finiteness property for all m > M j . Fix such an m and its sign δ too. Denote by D = D F , p j (δm) the digit set corresponding to F and p j (δm). Notice that D ⊂ Z is a complete residue system modulo p j (δm).
If 1, α, . . . , α l−1 is a power integral basis of O then by Győry's theorem there exist 1 ≤ j ≤ t, δ = ±1, m ∈ N such that α = α j − δm. We have seen in the last paragraph that all but finitely many m the numbers α j − δm together with the digit sets D form a number system with finiteness property.
Finally for each of the finitely many remaining values of m one can decide algorithmically the finiteness property by Theorem 2.9. is a finite extension field of K. Győry [16] proved that if U is a ring and a free O-module in L, then it admits finitely many classes of O-power integral bases. A representative of each class is effectively computable. Each class is closed under translation by elements of O. To generalize Theorem 6.2 to this situation would require the generalization of Remark 4.4 to all m ∈ O, such that all conjugates of m are large enough. We have no idea how to prove such a result.