Sep 24, 2002 - 1 = 0 < 1 < 2 < ··· to obtain representations of positive integers. ... different from N are the so-called canonical number systems (for short CNS).
Scheicher, K. and Thuswaldner, J.M. Osaka J. Math. 41 (2004), 327–351
ON THE CHARACTERIZATION OF CANONICAL NUMBER SYSTEMS ¨ KLAUS SCHEICHER and JORG M. THUSWALDNER
(Received September 24, 2002) 1.
Introduction
It is well known that each positive integer can be expressed uniquely as a sum = 0 + 1 + ··· + with an integral base number ≥ 2, 6 = 0 and ∈ {0 . . . − 1}. This concept can be generalized in several directions. On the one hand the base sequence 1, , 2 , . . . can be replaced by a sequence 1 = 0 < 1 < 2 < · · · to obtain representations of positive integers. Of special interest is the case where the sequence { }∞ =0 is defined by a linear recurrence. A famous example belonging to this class is the so-called Zeckendorf representation. On the other hand, one can generalize the set of numbers which can be represented. We mention two kinds of number systems belonging to this class: The so called β-expansions introduced by R´enyi [27] which are representations of real numbers in the unit interval as sums of powers of a real base number β. These digit representations of real numbers are strongly related to digit representations of positive integers if β is a zero of the characteristic polynomial of a linear recurring base sequence { }∞ =0 . Of special interest is the case where β is a Pisot number. These expansions have been extensively studied. We mention here the papers BerendFrougny [6] , Frougny [12, 13], Frougny-Solomyak [14, 15] and Loraud [25] and refer to the references given there. Another kind of number systems which admit the representation of a set which is different from N are the so-called canonical number systems (for short CNS). Since CNS form the main object studied in the present paper we recall their definition (cf. Akiyama-Peth˝o [2]). DEFINITION 1.1.
Let ( ) :=
be such that
≥ 1 and
= 1 (set
+
−1
−1
= 0 for
+ ···+
0
∈ Z[ ]
> ). Let N = {0 1 . . . | 0 | − 1} and
The first author was supported by the Austrian Science Foundation Project S8305. The second author was supported by the Austrian Science Foundation Project P-14200-MAT.
328
K. SCHEICHER
AND
J.M. THUSWALDNER
R be the quotient ring R = Z[ ]/ ( )Z[ ] • We say that γ ∈ R has a finite representation if it admits a representation of the shape γ=
0
+
1
+ ···+
with ∈ N for 0 ≤ ≤ and 6 = 0 for 6 = 0 (set = 0 for > ). • The numbers = (γ), ≥ 0, are called the digits of γ with respect to ( ( ) N ). • The pair ( ( ) N ) is called canonical number system or CNS in R, if each γ ∈ R has a finite representation. N is called digit set of this CNS. • If ( ) is irreducible, then let α be one of its zeros. In this case R is isomorphic to Z[α], the ring generated by Z and α. Therefore we may replace by α in the above expansions. In this case, we simplify the notation ( ( ) N ) to (α N ) and α is called base of this CNS. Unlike for “ordinary” number systems, where it is clear that each integer ≥ 2 can serve as base, it is a difficult problem to determine which polynomials provide CNS. Despite there are many papers dealing with the characterization of possible polynomials — we will give a detailed overview in the next section — there does not exist a complete characterization up to now. Some of the known results provide the characterization of the polynomials for some classes of CNS, others give algorithms that allow to decide whether a given polynomial ( ) forms a CNS or not. In the present paper we have two aims. First we want to present a fast algorithm for deciding whether a given ( ) is a CNS polynomial or not, in a second step we use this new algorithm to characterize a large class of CNS polynomials. Our results prove Conjecture 1 of Akiyama-Peth˝o [2] and provide a conditional proof of Conjecture 2 of the same paper. The paper is organized as follows. In the next section we discuss earlier results on the characterization of CNS and state some easy facts about them. Section 3 is devoted to the definition of certain graphs and automata which reflect many important properties of CNS and thus are very important for the proofs of our results. In Section 4 we establish a fast algorithm which allows to decide whether a given ( ) is a CNS polynomial or not (Theorem 4.4). In Section 5 we use our algorithm to characterize a large class of CNS polynomials (Theorem 5.8). Section 6 contains characterizations of cubic (Theorem 6.1) and quartic (Theorem 6.2) CNS under a certain condition. In Section 7 some numerically found examples are presented dealing with conjectures on CNS polynomials. We finish the paper with some remarks in Section 8.
CHARACTERIZATION
2.
OF
CANONICAL NUMBER SYSTEMS
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Some facts about canonical number systems
In this section we want to review some earlier results on CNS. Some of them will be used in the proofs of our results. As mentioned in the introduction it is an open problem to give a complete characterization of all polynomials that provide a CNS. Nevertheless, there are many partial results. Knuth [21] considered certain examples of bases, one of them the Gaussian integer −1 + which is intimately related to the famous “twin dragon” fractal (as for the connection between fractals and CNS cf. also Akiyama-Thuswaldner [4, 5], K´atai [17] and Scheicher-Thuswaldner [29]). The first systematic treatment was given in K´ataiSzab´o [20], where all Gaussian integers which are CNS bases are characterized. This result was generalized to quadratic integers in K´atai-Kov´acs [18, 19] and independently in Gilbert [16]. K˝ornyei [22] dealt with a special class of cubic integers and very recently Brunotte [9, 10] characterized all CNS whose bases are roots of trinomials. As for the general case Kov´acs [23] proved that an algebraic integer α gives rise to a CNS if its minimal polynomial ( ) = + −1 −1 + · · · + 1 + 0 satisfies 2≤
0
≥
1
≥ ··· ≥
−1
≥1
However, this characterization is not complete. Examples not contained in this class are provided in Kov´acs-Peth˝o [24], where also an algorithm is established that decides whether a given α is a CNS base or not. Recently, Akiyama-Peth˝o [2] developed a much faster algorithm than the one in [24]. In particular, they proved the following result. Let be a positive integer and α be an algebraic integer of degree with P | | then it minimal polynomial ( ) as in Definition 1.1. If 0 ≥ (1 + −1 ) =1 suffices to check for (2 ) elements of R whether they have finite representation in order to decide if ( ) is a CNS polynomial or not. These (2 ) numbers are given explicitly. Since this criterion works for all polynomials satisfying (2.1)
X =1
| |
0 is a certain computable constant. The finiteness of implies that the sequence { ( )} ≥0 becomes ultimately periodic for each ∈ R. Let P be the set of all points which can occur in such a period, i.e. P := { ∈ It is clear from the definition of tation of the shape (2.2)
| ∃ω ∈ N :
ω
=
and P that each
=
X
+
( )}
∈ R admits a unique represen-
+1
=0
with ∈ P, ∈ N and ∈ N as small as possible. Since has a representation of the shape (2.3)
=
ω−1 X
+
modω
is a periodic point, it
ω
=0
for each
∈ N. For this reason we will use the abbreviation = ([
ω−1 · · · 0 ]
∞
···
0)
for the representation (2.2). Here [ ω−1 · · · 0 ]∞ is the infinite repetition of the string ∞ · · · 0 ) we will simply write ( · · · 0 ). As mentioned ω−1 · · · 0 . Instead of ([0] in Definition 1.1 in this case the representation is called finite. Generalizing Definition 1.1 we will use the notation ( ) for the -th digit in the representation of , i.e. the coefficient of in this representation.
CHARACTERIZATION
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CANONICAL NUMBER SYSTEMS
The following lemma contains an easy criterion for a CNS.
331
( ) to be the polynomial of
Lemma 2.1. Suppose that each zero of ( ) has modulus greater than 1 and N is a complete residue system mod( ( )). Then ( ( ) N ) is a CNS if and only if P = {0}. Proof. If P = {0} then (2.2) ensures that ( ( ) N ) is a CNS. If 0 6 = ∈ P and ( ) = then has a representation of the shape (2.3). If all (0 ≤ ≤ ω − 1) were equal to zero this would imply = 0, a contradiction. Since the representation of is ([ ω−1 · · · 0 ]∞ ) this representation can not be finite. ω
We want to mention here that γ ∈ R has finite representation if and only if the sequence { (γ)} ≥0 is ultimately zero. This is an easy consequence of the definition of . In the remaining part of the paper we will always assume that N = {0 1 . . . | 0 | − 1} is a complete residue system mod( ( )) and that all zeros of ( ) are greater than 1 in modulus. By the above considerations this ensures that each γ ∈ R admits a representation of the shape (2.2). 3.
Definition of graphs and automata
In this section we want to define certain classes of directed graphs. These graphs will be used to perform the addition of fixed numbers on the space of representations. Furthermore, we will state some properties of these graphs and discuss their relation to so-called transducer automata. First of all we want to give a definition of this kind of automata. DEFINITION 3.1 (cf. Berstel [8] or Eilenberg [11]). The 6-tuple A=( ) is called a finite state transducer automaton if 0 • , and are nonempty, finite sets, and • : × → and : × → are unique mappings. The sets and are called input and output alphabet, respectively. is called the set of states and 0 is the starting state. The mappings and are called transition and result function, respectively. A finite automaton works as follows. The automaton starts at time 0 at the state . At each discrete time , the automaton reads an input digit and determines the 0 corresponding output digit ′ = ( ) as well as the next state +1 = ( ). A transducer automaton can be interpreted as a labeled directed graph in the following way. The vertices of are the elements of the set of states . Furthermore, there exists an edge from a vertex
1
to a vertex
2
labeled by
1
|
′ 1,
1|
1
′
1 −−→
2
for
332
short, if ( 1 1 ) = the present paper.
K. SCHEICHER 2
and
(
1
1)
AND
J.M. THUSWALDNER
= 1′ . We will need this interpretation frequently in
DEFINITION 3.2. Let ( ) be as in Definition 1.1. We say that the addition of the number ∈ R is computable by the finite state transducer ( ), if • for any γ ∈ R the transducer ( ) is able to read the digits (γ) of the representation of γ as input string and returns the digits (γ + ) of the representation of γ + as output string. • a finite representation of γ results in a finite representation of γ + . The transducer (1) is called the counting automaton or adding machine of ( ( ) N ). We will need the transducers ( ) for certain numbers ∈ R in order to derive our characterization results. These transducers will emerge from the following infinite labeled directed graph. Let A(R) be the labeled directed graph with set of vertices R. The edges connecting two vertices are defined as follows. Let 0 , 1 be two vertices of R. Then there exists an edge from 0 to 1 labeled by 0 | 0′ with 0 , 0′ ∈ N if and only if (3.1)
0
+
0
′ 0
=
+
1
This edge will be denoted by 0|
0
′
0 −−→
1
Of course, relation (3.1) can be iterated. If one starts at the vertex 0 and uses as input the digits of the representation of a number 0 + · · · + , one ends at the state +1 where (3.2)
0
+
0
+ ···+
=
′ 0
+ ···+
′
+1
+
+1
The and ′ can be interpreted as input and output digits, respectively. Suppose that we use the digits of the finite representation = ( · · · 0) as input digits (from right to left) starting at a vertex ∈ R. Then we can see from (3.2) that the sequence 0′ , 1′ , . . . of output digits is the representation + = [
′
′ +ω
···
′
+1 ]
∞ ′
′
···
′ 0
The representation formed from the output string is obviously always finite, i.e. 0 and ω = 1, if ( ( ) N ) is a CNS.
′
′ +1
=
CHARACTERIZATION
CANONICAL NUMBER SYSTEMS
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333
The above procedure becomes more clear if we set up a new type of representation for the elements of R. This representation has been found independently in Brunotte [9, 10]. Lemma 3.3. Let sentation
∈ R has a unique repre-
( ) be as in Definition 1.1. Each
(3.3)
=
−1 X =0
with (3.4)
=
X
ε
(ε ∈ Z
+
=1
= 0 ...
− 1)
For such sums we will use the notation = (ε1 . . . ε )ε
(3.5)
This representation will be called the ε-representation of . The change ( 0 ... −1 ) → (ε1 . . . ε )ε corresponds to a linear base transformation of the lattice R. Proof.
The equations (3.4) provide a linear system of
. ..
−1
1
0 ··· .. . .. . 2 ···
ε1 0 .. .. . . = . 0 .. ε
−1
.. . .. . 0
equations
This system has a unique solution such that ε ∈ Z for all , since ∈ Z. Let
= 1 and
,
= (ε1 . . . ε )ε . We will now examine how this ε-representation changes if |
′
we move along the edge −−→ ′ in A(R), i.e. we will determine the ε-representation of ′ from the ε-representation of . By the above lemma we have (3.6)
=
−1 X =0
X
ε
+
=1
By the definition of the edges of A(R) there exists a unique (3.7)
ε1
1
+ ···+ ε
+ =
0
+
′
∈ Z such that
334
K. SCHEICHER
J.M. THUSWALDNER
AND
Since + we can subtract
X
−
=0
′
+ = ′
+ ···+
0
=0
times this minimal polynomial from (3.6) to obtain =
Since
−1
−1
+ =
′
+
X
ε
+
=1
!
this implies that
−1 X =0
−
+1
+
X
ε
+ +1
=1
!
= (− ε1 . . . ε
−1 )ε
Thus the ε-representation of ′ emerges from the ε-representation of by canceling ε , shifting ε1 . . . ε −1 to the right and inserting − , which is defined according to (3.7), as the first element. DEFINITION 3.4. An edge of the shape (ε1 . . . ε )ε → ( ε1 . . . ε −1 )ε in A(R) is called an edge of type . If we emphasize on the type of an edge we will use the notation type
(ε1 . . . ε )ε −−−→ ( ε1 . . . ε
−1 )ε
We will be interested in subgraphs of A(R) which are closed in a certain sense. To this matter we need the following definition. DEFINITION 3.5. A number +1 is called reachable from 0 if there exist 0 . . . ∈ N such that (3.2) holds. We will denote this by 0 +1 . The series of states 0|
0
is called the path connecting
0
′
1|
0 −−→
1
with ={
′
+1 .
|
′
|
′
1 −−→ · · · −−→
Let ′
+1
⊂ R. Then ∈
denotes the set
}
In the following lemma we show the existence of a rather “small” set is closed in the sense of the above definition.
, which
Lemma 3.6. Let ( ) be as in Definition 1.1 and suppose that ( ) fulfills the AP condition (2.1). Then = { | = (ε1 . . . ε )ε ε ∈ {−1 0 1}} satisfies = . has 3 elements. Thus
CHARACTERIZATION
OF
CANONICAL NUMBER SYSTEMS
335 |
′
Proof. We have to show that for each ∈ the existence of an edge −−→ ′ in A(R) implies ′ ∈ . Suppose that = (ε1 . . . ε )ε with ε ∈ {−1 0 1}. From the AP condition it follows that X X | |< 0 ε ≤ =1
=1
and thus
− Since
0
< ε1
1
+ ···+ ε
