REAL NUMBERS HAVING ULTIMATELY PERIODIC ...
arXiv:cs/0212018v1 [cs.CC] 10 Dec 2002
REAL NUMBERS HAVING ULTIMATELY PERIODIC REPRESENTATIONS IN ABSTRACT NUMERATION SYSTEMS P. LECOMTE AND M. RIGOy Abstract. Using a genealogically ordered in nite regular language, we know how to represent an interval of R. Numbers having an ultimately periodic representation play a special role in classical numeration systems. The aim of this paper is to characterize the numbers having an ultimately periodic representation in generalized systems built on a regular language. The syntactical properties of these words are also investigated. Finally, we show the equivalence of the classical -expansions with our generalized representations in some special case related to a Pisot number .
1. Introduction Enumerating the words of an in nite regular language L over a totally ordered alphabet ( ; 1. Therefore the problem of relating this number to the set of numbers having an ultimately periodic representation clearly appears in the case of abstract systems. This paper has the following organization. First we recall de nitions and notation about abstract numeration systems and the representation of real numbers. Next we recall the general assumptions we consider when dealing with the representation of real numbers. As stated before, these assumptions are related to the asymptotic behavior of the counting functions of the languages accepted from the di erent states of the minimal automaton of L. The reader could already note that we have slightly simpli ed the presentation given in [11]. The aim of Sections 2 and 3 is to give a summary of the relevant facts given in [10, 11, 15]. In Section 4 we study the syntactical properties of the ultimately periodic representations. We show that the corresponding language of in nite words is !-rational. This section has an automata theory avor and can be read separately from the rest of the paper. In Section 5, we obtain formulas for computing e ectively the numerical value of an ultimately periodic representation. Moreover we show that the language made up of the ultimately periodic representations is dense in the set of all the representations. In [11], it is explained that for an abstract system built upon an arbitrary regular language L, a real number can have one, a nite number or even an in nite number of representations and the situation can be completely determined from the language L (actually, from the asymptotic behavior of the counting functions associated to the di erent states). In Section 6, we show how to modify the language L to obtain a new numeration system having exactly the same representations except that in this new system a number has at most two representations. Roughly speaking, we remove from the minimal automaton of L the useless states which are giving redundant representations. In Section 7, we use some intervals Iw (a real number x belongs to Iw if x has a representation having w as pre x) to obtain a characterization of the real numbers having an ultimately periodic representation. From the ideas given in this result and its proof, we derive two algorithms for computing the representation of an arbitrary real number. These algorithms can be viewed as a generalization of the greedy algorithm used to compute -expansions [14] and rely on the use of some a ne functions completely de ned by the minimal automaton of the language. We also present a dynamical system built upon those a ne functions, the points having an ultimately periodic orbit in this dynamical system being exactly the real numbers having an ultimately periodic representation (this system is a generalization of the intervals exchange transformation [4, 12]). In Section 8, thanks to our algorithm of representation, we obtain another characterization of the real numbers having an ultimately periodic representation, these numbers are the xed points of composition of some a ne functions. Moreover, this composition can actually be viewed as a word belonging to a regular language over a nite alphabet of functions. In the last section, we consider a Pisot number . To this number, corresponds a unique linear Bertrand numeration system [2]. If L is the language of representations of the integers in this latter Bertrand system then the representations of the real numbers in the abstract system built upon L and the classical -developments are the same. So thanks to a famous result of Klaus Schmidt, in this particular
REAL NUMBERS HAVING ULTIMATELY PERIODIC REPRESENTATIONS. . .
3
case, we know precisely the structure of the set of real numbers having an ultimately periodic representation. This set is Q( ). 2. Preliminaries Let us precise notation and de nitions. Let be a nite alphabet. We denote by the free monoid generated by with identity ". Let L be an in nite regular language and ML = (Q; q0 ; ; ; F ) be its minimal automaton having Q as set of states, q0 as initial state, F as set of nal states. The transition function : Q ! Q of this automaton is naturally extended to Q and we often write q:w as a shorthand for (q; w), q 2 Q, w 2 . (For more about automata theory see for instance [3].) If q 2 Q, we denote by Lq the language accepted in ML from the state q, i.e., Lq = fw 2 j q:w 2 F g: In particular, Lq0 = L. In this paper, we shall extensively use the following linear recurrent sequences de ned for each q 2 Q by uq (n) = #(Lq \
n
);
vq (n) = #(Lq \
n
):
Since the initial state q0 plays a special role, if q = q0 then we simply write u(n) and v(n) (in the literature, u(n) is often called the growth function, the counting function or even the complexity of the language L). Let ( ; 0 there exist N such that w and wN have a common pre x of length ‘ and jwN j ‘ + #Q. When reading the su x of length #Q of wN in ML we go at least twice through the same state q and let u be the corresponding factor of wN such that q:u = q. Therefore, wN can be written xuz with jxj l and it is clear that xun z 2 L for all n 1. So xu! 2 L1 and has a pre x of length ‘ in common with w. We can therefore build a sequence of ultimately periodic words converging to w. 6. Simplifying the language In the rst part of this section, we explain how a real number can have more than one representation and even an in nite number of representations. Next, we explain, how we can slightly change the language to avoid this situation of having an in nite number of representations but without altering the other representations. In [11], we gave a partition of the interval [1= ; 1] into intervals Iw . These intervals will play a central role in what follows so let us recall their de nition. First consider the k-ary system. In this system, the representation of a real number x 2 [1=10; 1] has a pre x w = w0 (w0 6= 0) if x belongs to the interval n w # "P Pn 1 n 1 n i 1 1 + i=0 wi k n i 1 i=0 wi k ; Iw = kn kn Observe that the endpoints of the intervals Iw are the only numbers having two representations. For instance, if k = 10 then 2=10 can be written 0; 1999 and 2=10 is the upper bound of I1 but it can also be written 0; 2000 and is the lower bound of I2 . For an abstract numeration system, we have the following de nition. De nition 18. A real number x 2 [1= ; 1] belongs to Iw if there exist a representation of x having w as pre x. For an arbitrary regular language L, the set L1 can contains an in nite number of words having w as pre x even if the length of the interval Iw is zero. In this situation, all the elements of L1 having w as pre x are representing the same real number x and Iw = [x; x]. Therefore x has an in nite number of representations. To avoid this situation, we proceed as follows. We can only consider the states q such that aq > 0. We have the following rules If aq 6= 0 and there exists w such that p:w = q, then ap 6= 0. If aq = 0 and there exists w such that q:w = p, then ap = 0. Indeed, in the rst case, if p:w = q then un (p) un jwj (q). In the second case, if q:w = p then un (q) un jwj (p). Hence we obtain the conclusion by dividing both sides by Pq0 (n) n . We can split the set of states of ML into two subsets Q0 = fq j aq = 0g and Q>0 = fq j aq > 0g. If we consider only the states of Q>0 and the corresponding edges connecting those states, we obtain a new automaton accepting a new language L0 . Representations of real numbers for the numeration system built upon L or L0 are the same except that for the system built on L0 a real number has at most two representations (only when it is the endpoint of some interval Iw ). Example 19. We consider the language L accepted by the automaton depicted in Figure 5. This language is such that the number of words beginning with b (resp. a or c) has a polynomial (resp. exponential) behavior. This means that the length of the interval Ib is zero (computations are given in [11, Example 6]). Therefore the greatest word in the lexicographical ordering of L1 beginning with a represents the same real number x as any word in L1 beginning with b or the smallest word
REAL NUMBERS HAVING ULTIMATELY PERIODIC REPRESENTATIONS. . .
a
b
a Q
b
a b
a,c
0
Q
11
b
>0
Figure 5. A trim minimal automaton. beginning with c. Removing the states of Q0 , gives a new language L0 . In the numeration system built upon L0 , the number x has exactly two representations: the greatest word beginning with a and the smallest one beginning with c. In this latter system, if w is pre x of an in nite number of words in L then the length of the interval Iw is strictly positive. In the following of this paper, we shall assume that aq > 0 for all states q in ML except possibly for the sink state. 7. Determining the numbers having an ultimately periodic representation Let us have a closer look at those intervals Iw (for the details, the reader is referred to [11]). If w 2 ‘ is pre x of an in nite number of words in L then the interval Iw is given by (4) X uq :m (n ‘) X uq :m (n ‘) v(n 1) v(n 1) 0 0 + + lim ; lim n!1 n!1 v(n) v(n) v(n) v(n) ‘ ‘ m2 m<w
m2 m w
Using the fact that (see [11, Proposition 5]) limn!1 vq (n)=v(n) = aq limn!1 uq (n)=vq (n) = (
(5)
the interval Iw be rewritten as (6)
1
+
1 X
‘+1
m2 ‘ m<w
aq0 :m ;
1
+
1)= 1 X
‘+1
aq0 :m :
m2 ‘ m w
Notice that this formulation di ers slightly from [11] because we have here aq0 = 1 and the others aq ’s are strictly positive (except for the sink). Observe also that the length of Iw is ‘+11 aq0 :w > 0. Remark 20. Notice that if w 2 is pre x of an in nite number of words in L then q0 :w is a coaccessible state (so it cannot be the sink) and with our assumptions, aq0 :w > 0. Example 21. Consider the numeration system associated to the language accepted by the automaton depicted in Figure 6. Here, easy computations show that = 2, aq0 = aq2 = 1 and aq1 = 2 (to obtain the aq ’s, one has only to compute the eigenvectors of the eigenvalue of the adjacency matrix). Any word in L1 begins with a, so Ia = [1=2; 1] (instead of this reasoning, formula (6) could also be used to compute the values of the endpoints of Ia ). This interval is partitioned into three parts, Iaa = [1=2; 5=8]; Iab = [5=8; 3=4]; Iac = [3=4; 1]:
12
P. LECOMTE AND M. RIGO
c a
a 0
b
1
2
c
Figure 6. A trim minimal automaton. Thus if a real number x belongs to Ia then we have an in nite word representing x beginning with a , 2 . For the next step, we have Iaac = Iaa = [1=2; 5=8];
Iaba = Iab = [5=8; 3=4]
and Iac is split into three parts Iaca = [3=4; 5=6];
Iacb = [5=6; 7=8];
Iaca = [7=8; 1]:
Actually the form of the partition of an interval Iw into intervals Iw depends only on the state q0 :w and not on the word w itself. De nition 22. If 0
< f[
1, the strictly increasing function ; ]
: [ ; ] ! [0; 1] : x 7!
x
maps the interval [ ; ] onto [0; 1]. If z belongs to [ ; ] then we say that f[ ; ] (z) is the relative position of z inside [ ; ]. We denote by Lw (resp. Uw ) the lower (resp. upper) bound of the interval Iw . Roughly speaking, the next proposition states that two intervals corresponding to the same state are homothetic. But rst, we need a technical lemma. Lemma 23. We have, for all states q, X aq: = in particular,
aq
2
P 2
a q0 : = .
Proof. Clearly, Iw = [ 2 Iw and the length of Iw is equal to conclusion follows directly from (6).
P 2
jIw j. The
Proposition 24. Let m and w be two words such that q0 :m = q0 :w. For all 2 , the interval Im exists1 i Iw exists and the relative position of Lm (resp. Um ) inside Im is equal to the relative position of Lw (resp. Uw ) inside Iw . Proof. The interval Im exists if aq0 :m > 0. Since q0 :m = q0 :w then for all 2 , aq0 :m > 0 i aq0 :w > 0. Using (6), the relative position of Lm inside Im is given by 1 1 X 1 Lm aq0 :u = jmj+1 aq0 :m jmj+1 u2 jmj u<m
this can be rewritten as jmj
(
1) aq0 :m
( Lm
1)
X u2 jmj u<m
aq0 :u aq0 :m
1An interval I exists if there exists a word in L w 1 having w as pre x. This means that w is pre x of an in nite number of words in L.
REAL NUMBERS HAVING ULTIMATELY PERIODIC REPRESENTATIONS. . .
and using the de nition of Lm , we get 1 X 1 aq0 :u (7) aq0 :m jmj+1
X
13
aq0 :u :
u2 jmj u<m
u2 u<m
First notice that the sum over the words u of length jmj + 1 and lexicographically less than m can be split into two subsets: the words m with < and the words having a pre x of length jmj lexicographically less than m. So (7) can be written X X 1 1 X aq0 :m + aq0 :u aq0 :u aq0 :m jmj+1 jmj 2
< fA0q1 ;a : fA0q ;b : [1=4; 1=2] ! [0; 1] : x 7! 4x 1; > : f A0 1 : [1=2; 1] ! [0; 1] : x 7! 2x 1: q1 ;c The automaton FL is depicted in Figure 7. For the sake of simplicity, fA0q1 ; is denoted by f , for = a; b; c (since it does not lead to any confusion). We also put an index a or c to id to remember the corresponding letter. A path f1 t f
fc
ida Aq
fb
Aq
0
fa id c
1
Aq
2
Figure 7. The automaton FL . in FL corresponds to the composition of a ne functions ft 1 inf reversed order. Through FL , we can determine the real numbers having ultimately periodic representations. Indeed, if we consider a cycle f1 f FL starting in Aq0 then, in view of t in Algorithm 30, if the unique xed point of the corresponding function F = ft f 1 is x then fI 1 (x), with fI : x 7! x 11 and fI 1 (y) = ( 1)y+1 , has an ultimately periodic representation (we are back in the initial state q0 and since x = F (x), we have the same initial value; due to the initialization step in Algorithm 30, we have to apply fI 1 once). As an example, the xed points of fb ida , fb idc fa ida and fb fc idc fa ida are respectively 1=3, 1=15 and 5=31 and therefore 2=3, 8=15 and 18=31 have ultimately periodic representations. From the path in F L , we also know these representations: (ab)! , (aacb)! and (aaccb)! . We can also consider a cycle f1 f FL starting in Aq1 instead of Aq0 and a t in to A . Once again, let x be the xed point of F = ft path g1 from g A s q0 q1 1. f 1 From Algorithm 30, the number fI 1 g1 1 g (x) has an ultimately periodic s representation. As an example, consider F = fc idc fa fc having 3=5 has xed point. A trivial path from Aq0 to Aq1 is given by ida , so fI 1 (3=5) = 4=5 has an ultimately periodic representation: a(cacc)! . Another path in FL from Aq0 to Aq1 is (ida ; fa ; idc ), so fI 1 fa 1 (3=5) = 23=40 is represented by aac(cacc)! . Let us introduce some notation. Let q
= fw j
F (Aq ; w)
= Aq g;
REAL NUMBERS HAVING ULTIMATELY PERIODIC REPRESENTATIONS. . .
17
where F is the transition function of FL . If w = f1 f to q , we denote t belongs by Fq;w the composed function (in reversed order) ft corresponding f to w. 1 Let q = fw j F (Aq0 ; w) = Aq g; If w = f1 f to q , we denote by (Fq;w ) 1 the composition of the inverse s belongs 1 1 functions f1 corresponding f to w. s Theorem 34. Let L be a regular language satisfying our basic assumptions. Set ( 1) y + 1 : fI 1 : y 7! The set of real numbers having an ultimately periodic representation is given by ffI
1
(Fq;w )
1
(x) j 9q 2 Q; z 2
q; w
2
q
: x = Fq;z (x)g:
Proof. This is a direct consequence of Algorithm 30. Remark 35. For all states q, the languages functions are regular.
q
and
q
over a
nite alphabet of
9. Equivalence with -development Let > 1 be a Pisot number. To this number corresponds a unique positional and linear Bertrand number system U = (Un )n2N having its characteristic polynomial equal to the minimal polynomial of [2]. We denote by L the language U (N) of all the normalized representations computed by the greedy algorithm (without leading zeroes) [5]. In this section we show that this latter language L satis es the hypotheses given in Section 3. We also prove that the representations of real numbers in the abstract numeration system built upon L and the classical -developments of the numbers in [ 1 ; 1] coincide. (For a presentation of the -development, we refer the reader to [13, Chapter 7] or [14].) De nition 36. Recall that a positional numeration system U = (Un )n2N is said to be a Bertrand numeration system if 8n 2 N; w 0n 2
U (N)
,w2
U (N):
As an example, the k-ary number system is a Bertrand system. p
Example 37. The golden ration = 1+2 5 is a Pisot number, indeed its minimal polynomial is P (X) = X 2 X 1 and the other root of P has modulus less than one. The polynomial P is also the characteristic polynomial of the linear recurrence relation de ned by Un+2 = Un+1 + Un ; n 2 N: If we consider the initial conditions U0 = 1 and U1 = 2, then as a consequence of the greedy algorithm, the set of representations of the integers is U (N) = f"g [ 1f0; 01g . Due to the particular form of the language U (N), it is clear that this system (namely the Fibonacci system) is the linear Bertrand number system associated to . Consider an arbitrary Pisot number . It is well known that the -development of one is nite or ultimately periodic [17]. In the rst case, e(1) = t1 t m and we de ne, as usual, 1))! : e (1) = (t1 m t1 (tm It is clear that we still have t2 tm 1 t1 t1 + 2+ + m + m+1 + : 1= Let > 1 be a real number. The set D of -developments of numbers in [0; 1) is characterized as follows.
18
P. LECOMTE AND M. RIGO
Theorem 38. [14] Let > 1 be a real number. A sequence (xn )n 1 belongs to D if and only if for all i 2 N, the shifted sequence (xn+i )n 1 is lexicographically less than the sequence e (1) or e (1) whenever e (1) is nite. For any real number > 1, we denote by F (D ), the set of nite factors of the sequences in D . Bertrand numeration systems are characterized by the theorem of Bertrand given below. Notice that U is not necessarily linear. Theorem 39. [1] Let U = (Un )n2N be a positional numeration system. Then U is a Bertrand numeration system if and only if there exists a real number > 1 such that 0 U (N) = F (D ). In this case, if e (1) = (dn )n 1 (or e (1) = (dn )n 1 whenever e (1) is nite) then U0 = 1 and Un = d 1 Un
1
+ d 2 Un
+
2
+ d0 + 1; n nU
1:
Let > 1 be a Pisot number. First we assume that e (1) is ultimately periodic; there exist minimal integers N 0, p 1 such that e (1) = t1
(t t N +1
N
t) N +p
!
:
The Bertrand numeration system U = (Un )n2N belonging to the class of positional systems related to is a linear numeration system satisfying the recurrence relation Un
= t 1 Un
1
+(tp+1
+
p+ 1t Un p+1
t1 ) Un
p 1
+ (tp + 1) Un
+
+ (t N +p
p
tN ) Un
N p;
n
N + p:
(In other words, (Un )n2N satis es the canonical beta polynomial of [9].) In what follows, is given and we denote U simply by U . The main point is the following. Since is a Pisot number, the set F (D ) = 0 U (N) is recognizable by a nite automaton A [6] (i.e., the -shift is so c). This automaton has N + p states q1 ; : : : ; qN +p . For each i 2 f1; : : : ; N + pg, there are edges labeled by 0; 1; : : : ; ti 1 from qi to q1 , and an edge labeled ti from qi to qi+1 if i < N + p. Finally, there is an edge labeled tN +p from qN +p to qN +1 . All states are nal and q1 is the initial state. The set F (D ) is recognized by the automaton depicted in Figure 8 (the sink is not represented).
0,...,t1−1 q1
t1
0,...,tN −1
0,...,t 2−1 t2 q2
t N−1
0,...,tN+p−1 qN+p
tN
t N+p t N+p−1
0,...,t
N+1
qN
t N+1
qN+1
−1
Figure 8. Automaton recognizing F (D ) = 0
U (N).
In an abstract numeration system, allowing leading zeroes changes the representations (indeed, 0w is genealogically greater than w then valS (0w) > valS (w), see for instance [10, Example 1]). Therefore, we modify slightly the automaton A to
REAL NUMBERS HAVING ULTIMATELY PERIODIC REPRESENTATIONS. . .
19
obtain an automaton A0 recognizing exactly U (N) (i.e., without leading zeroes). To that end, we add a new state q0 . There are edges labeled by 1; : : : ; t1 1 from q0 to q1 and an edge labeled t1 from q0 to q2 . This state q0 is the initial state of A0 and is also nal. The automaton A0 is sketched in Figure 9.
q
0
1,...,t1−1 0,...,t1−1
t1 0,...,t 2−1 t1
q1
q2 A
Figure 9. The automaton A0 recognizing
U (N).
So we consider the abstract system built upon L = U (N). To be able to compute the intervals Iw related to this system with formula (4), our task is now to determine the di erent sequences uqi (n). To that end, we use the speci c form of A0 . The rst word of length n + 1 in U (N) is 1(0)n and its numerical value is Un . In the same way, 1(0)n 1 is the rst word of length n. Therefore, (9)
uq0 (n) = Un
Un
1:
0
Since q0 is the initial state of A , as usual we write u(n) and v(n) instead of uq0 (n) and vq0 (n). Since is a Pisot number and the characteristic polynomial of U is the minimal polynomial of , there exists a real number such that n
Un For n
:
1, from the form of A0 we deduce that u(n) = (t1
1) uq1 (n
1) + uq2 (n
1)
and (10)
uq1 (n) = t1 uq1 (n
1) + uq2 (n
1) = u(n) + uq1 (n
As a consequence of (10), since all the states are uq1 (n) =
n X
1):
nal uq1 (0) = u(0) = 1, we
nd
u(i) + uq1 (0) = v(n):
i=1
From (10) we also have uq2 (n 1) = uq1 (n) t1 uq1 (n 1) and thus uq2 (n) = v(n + 1) t1 v(n). But considering the path in A0 , we get uq2 (n) = t2 uq1 (n 1) + uq3 (n 1). So we nd uq3 (n) = v(n + 2) t1 v(n + 1) t2 v(n). Continuing this way, for i N + p uqi (n) = v(n + i
1)
t1 v(n + i
2)
t2 v(n + i
3)
i 1tv(n):
We are now able to determine the endpoints of the intervals Iw . It is clear from n (9) that v(n) = Un . Since v(n 1) = v(n) u(n) using (5), it is clear that
20
v(n
P. LECOMTE AND M. RIGO
1)=v(n) ! 1= if n ! 1 and therefore, for i 2 N lim
n!1
v(n i) v(n uq1 (n i) = lim n!1 v(n v(n) i + 1) v(n
v(n 1) = v(n)
i + 1) i + 2)
i
:
In the same manner, lim
n!1
Continuing this way, for j
uq2 (n i) = v(n)
1 i
:
N + p and i 2 N
uqj (n i) = n!1 v(n) lim
j i 1
t1
j i 2
j 1t
We can now compute the di erent intervals Iw . The 1; : : : ; (t1
i
t1
1); t1 ; 10; : : : ; 1t1 ; 20; : : : ; (t1
i
:
rst words in
U (N)
are
1)t1 ; t1 0; : : : ; t1 t2 ; 100 : : :
Using (4) we have the intervals corresponding to words of length one j j j j+1 uq1 (n 1) Ij = [ ; + lim ]=[ ; ]; 1 n!1 v(n)
j < t1
and
t1 t1 t1 uq2 (n 1) ; + lim ] = [ ; 1]: n!1 v(n) For the words of length two, if 1 j < t1 and 0 k < t1 then I t1 = [
k j k uq1 (n 2) j k j k+1 j ] = [ + 2; + 2 ] Ijk = [ + 2 ; + 2 + lim n!1 v(n) and
j t1 j t1 uq2 (n 2) j t1 j + 1 ] = [ + 2; ]: Ijt1 = [ + 2 ; + 2 + lim n!1 v(n) For the words of length two beginning with t1 , we have I t1 j = [ and
t1
+
j t1 j+1 ; + 2 ]; 0 2
j < t2
uq3 (n 2) t2 t1 t2 t1 t2 ; + 2 + lim ] = [ + 2 ; 1]: 2 n!1 v(n) Continuing this way, it is straightforward computation to see that we have three situations: (1) if w = w0 w qi = q0 :w0 r with r w 1 and wr < ti then I t1 t2 = [
t1
+
Iw = [
r X
wi
i 1
r X
;
i=0
wi
i 1
+
r 1
]
i=0
(2) if w = w0 wr+1 w is such that qi = q0 :w0 rw r+s r w 1 , wr < ti and wr+1 w is the maximal word read from q :w w in A0 (in other r+s 0 0 r words, q0 :w0 w q1 and wr+1 w is the pre x of length s of e (1)) r = r+s then r r+s X X wi i 1 + r 1 ]: wi i 1 ; Iw = [ i=0
i=0
(3)
nally, if w = w0
r
is w a pre x of e (1) then
Iw = [
r X i=0
wi
i 1
; 1]:
REAL NUMBERS HAVING ULTIMATELY PERIODIC REPRESENTATIONS. . .
21
Now instead of considering the abstract numeration system built upon L = (N), we can consider the classical -development of a real number x 2 [1= ; 1]. U The rst digit of e (x) is an integer j belonging to f1; 2; : : : ; t1 g. Since -developments are computed through the greedy algorithm, it is clear that the rst digit is j < t1 if and only if x 2 Ij0 = [j= ; (j + 1)= ) and it is t1 if and only if x 2 It01 = [t1 = ; 1]. So the interval Ij for the abstract numeration systems considered above and the intervals Ij0 corresponding to the greedy algorithm are the same for the rst step (except that in the abstract system, a real number can have two representations but we can avoid this ambiguity by considering intervals of the form [a; b) and therefore the two intervals Ij and Ij0 will coincide exactly). By application of the greedy algorithm, we can compute intervals Iw0 such that x belongs to Iw0 if e (x) has w as pre x. Clearly those intervals Iw0 coincide with the intervals Iw and therefore, the classical -developments are the same as the representation obtained in the framework of the abstract numeration systems (naturally, under the extra assumptions of this section corresponding to regular languages associated to Pisot number). Now we can use a result of Klaus Schmidt concerning ultimately periodic developments [17] and state the following result. Theorem 40. If L is the language of all the representations of the integers in a linear Bertrand numeration system associated to a Pisot number then the set of real numbers having an ultimately periodic representation in the abstract system built upon L is exactly Q( ) \ [1= ; 1]: Remark 41. In this section, we have only considered the case e (1) ultimately periodic. If e (1) = t1 t nite (tm 6= 0) then the same situation holds. The m is construction of the automaton A is the same as before but with N = m and p = 0. All the edges from qm lead to q1 and are labeled by 0; : : : ; tm 1. The automaton A is depicted in Figure 10.
0,...,t1−1 q1
t1
0,...,t 2−1 t2 q2
0,...,t m −1 t m−1
Figure 10. The automaton A in the case e (1)
qm nite.
References [1] A. Bertrand-Mathis, Comment ecrire les nombres entiers dans une base qui n’est pas entiere, Acta Math. Acad. Sci. Hungar. 54 (1989), 237{241. [2] V. Bruyere, G. Hansel, Bertrand numeration systems and recognizability, Theoret. Comput. Sci. 181 (1997), 17{43. [3] S. Eilenberg, Automata, Languages and Machines, Vol. A, Academic Press, New York, (1974). [4] S. Ferenczi, C. Holton, L. Q. Zamboni, Structure of three interval exchange transformations. I. An arithmetic study, Ann. Inst. Fourier (Grenoble) 51 (2001), 861{901. [5] A. S. Fraenkel, Systems of numeration, Amer. Math. Monthly 92 (1985), 105{114. [6] C. Frougny, B. Solomyak, On representation of integers in linear numeration systems, in Ergodic theory of Z d actions (Warwick, 1993{1994), 345{368, London Math. Soc. Lecture Note Ser. 228, Cambridge University Press, Cambridge, (1996). [7] P. J. Grabner, P. Liardet, R. F. Tichy, Odometers and systems of numeration, Acta Arith. 70 (1995), 103{123. [8] P. J. Grabner, M. Rigo, Additive functions with respect to numeration systems on regular languages, to appear in Monatsh. Math.
22
P. LECOMTE AND M. RIGO
[9] M. Hollander, Greedy numeration systems and regularity, Theory Comput. Syst. 31 (1998), 111{133. [10] P. B. A. Lecomte, M. Rigo, Numeration systems on a regular language, Theory Comput. Syst. 34 (2001), 27{44. [11] P. Lecomte, M. Rigo, On the representation of real number using regular languages, Theory Comput. Systems 35 (2002), 13{38. [12] L.-M. Lopez, P. Narbel, Substitutions and interval exchange transformations of rotation class, Theoret. Comput. Sci. 255 (2001), 323{344. [13] M. Lothaire, Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications, 90. Cambridge University Press, Cambridge, (2002). [14] W. Parry, On the -expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401{416. [15] M. Rigo, Numeration systems on a regular language : Arithmetic operations, recognizability and formal power series, to appear in Theoret. Comput. Sci. (2001). [16] J. Shallit, Numeration systems, Linear recurrences, and Regular sets, Information and Computation 113 (1994), 331{347. [17] K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), 269{278. [18] W. Thomas, Automata on in nite objects, in Handbook of theoretical computer science, Vol. B, J. Van Leeuwen Ed., Elsevier, Amsterdam, (1990), 133{191. [19] S. Yu, Regular languages, in Handbook of formal languages, Vol. 1, 41{110, Springer, Berlin, (1997). E-mail address: [email protected] E-mail address: [email protected] (P. L. and M. R.) Institut de Mathematique, Universite de Liege, Grande Traverse 12 (B 37), B-4000 Liege, Belgium.
REAL NUMBERS HAVING ULTIMATELY PERIODIC REPRESENTATIONS IN ABSTRACT NUMERATION SYSTEMS P. LECOMTE AND M. RIGOy Abstract. Using a genealogically ordered in nite regular language, we know how to represent an interval of R. Numbers having an ultimately periodic representation play a special role in classical numeration systems. The aim of this paper is to characterize the numbers having an ultimately periodic representation in generalized systems built on a regular language. The syntactical properties of these words are also investigated. Finally, we show the equivalence of the classical -expansions with our generalized representations in some special case related to a Pisot number .
1. Introduction Enumerating the words of an in nite regular language L over a totally ordered alphabet ( ; 1. Therefore the problem of relating this number to the set of numbers having an ultimately periodic representation clearly appears in the case of abstract systems. This paper has the following organization. First we recall de nitions and notation about abstract numeration systems and the representation of real numbers. Next we recall the general assumptions we consider when dealing with the representation of real numbers. As stated before, these assumptions are related to the asymptotic behavior of the counting functions of the languages accepted from the di erent states of the minimal automaton of L. The reader could already note that we have slightly simpli ed the presentation given in [11]. The aim of Sections 2 and 3 is to give a summary of the relevant facts given in [10, 11, 15]. In Section 4 we study the syntactical properties of the ultimately periodic representations. We show that the corresponding language of in nite words is !-rational. This section has an automata theory avor and can be read separately from the rest of the paper. In Section 5, we obtain formulas for computing e ectively the numerical value of an ultimately periodic representation. Moreover we show that the language made up of the ultimately periodic representations is dense in the set of all the representations. In [11], it is explained that for an abstract system built upon an arbitrary regular language L, a real number can have one, a nite number or even an in nite number of representations and the situation can be completely determined from the language L (actually, from the asymptotic behavior of the counting functions associated to the di erent states). In Section 6, we show how to modify the language L to obtain a new numeration system having exactly the same representations except that in this new system a number has at most two representations. Roughly speaking, we remove from the minimal automaton of L the useless states which are giving redundant representations. In Section 7, we use some intervals Iw (a real number x belongs to Iw if x has a representation having w as pre x) to obtain a characterization of the real numbers having an ultimately periodic representation. From the ideas given in this result and its proof, we derive two algorithms for computing the representation of an arbitrary real number. These algorithms can be viewed as a generalization of the greedy algorithm used to compute -expansions [14] and rely on the use of some a ne functions completely de ned by the minimal automaton of the language. We also present a dynamical system built upon those a ne functions, the points having an ultimately periodic orbit in this dynamical system being exactly the real numbers having an ultimately periodic representation (this system is a generalization of the intervals exchange transformation [4, 12]). In Section 8, thanks to our algorithm of representation, we obtain another characterization of the real numbers having an ultimately periodic representation, these numbers are the xed points of composition of some a ne functions. Moreover, this composition can actually be viewed as a word belonging to a regular language over a nite alphabet of functions. In the last section, we consider a Pisot number . To this number, corresponds a unique linear Bertrand numeration system [2]. If L is the language of representations of the integers in this latter Bertrand system then the representations of the real numbers in the abstract system built upon L and the classical -developments are the same. So thanks to a famous result of Klaus Schmidt, in this particular
REAL NUMBERS HAVING ULTIMATELY PERIODIC REPRESENTATIONS. . .
3
case, we know precisely the structure of the set of real numbers having an ultimately periodic representation. This set is Q( ). 2. Preliminaries Let us precise notation and de nitions. Let be a nite alphabet. We denote by the free monoid generated by with identity ". Let L be an in nite regular language and ML = (Q; q0 ; ; ; F ) be its minimal automaton having Q as set of states, q0 as initial state, F as set of nal states. The transition function : Q ! Q of this automaton is naturally extended to Q and we often write q:w as a shorthand for (q; w), q 2 Q, w 2 . (For more about automata theory see for instance [3].) If q 2 Q, we denote by Lq the language accepted in ML from the state q, i.e., Lq = fw 2 j q:w 2 F g: In particular, Lq0 = L. In this paper, we shall extensively use the following linear recurrent sequences de ned for each q 2 Q by uq (n) = #(Lq \
n
);
vq (n) = #(Lq \
n
):
Since the initial state q0 plays a special role, if q = q0 then we simply write u(n) and v(n) (in the literature, u(n) is often called the growth function, the counting function or even the complexity of the language L). Let ( ; 0 there exist N such that w and wN have a common pre x of length ‘ and jwN j ‘ + #Q. When reading the su x of length #Q of wN in ML we go at least twice through the same state q and let u be the corresponding factor of wN such that q:u = q. Therefore, wN can be written xuz with jxj l and it is clear that xun z 2 L for all n 1. So xu! 2 L1 and has a pre x of length ‘ in common with w. We can therefore build a sequence of ultimately periodic words converging to w. 6. Simplifying the language In the rst part of this section, we explain how a real number can have more than one representation and even an in nite number of representations. Next, we explain, how we can slightly change the language to avoid this situation of having an in nite number of representations but without altering the other representations. In [11], we gave a partition of the interval [1= ; 1] into intervals Iw . These intervals will play a central role in what follows so let us recall their de nition. First consider the k-ary system. In this system, the representation of a real number x 2 [1=10; 1] has a pre x w = w0 (w0 6= 0) if x belongs to the interval n w # "P Pn 1 n 1 n i 1 1 + i=0 wi k n i 1 i=0 wi k ; Iw = kn kn Observe that the endpoints of the intervals Iw are the only numbers having two representations. For instance, if k = 10 then 2=10 can be written 0; 1999 and 2=10 is the upper bound of I1 but it can also be written 0; 2000 and is the lower bound of I2 . For an abstract numeration system, we have the following de nition. De nition 18. A real number x 2 [1= ; 1] belongs to Iw if there exist a representation of x having w as pre x. For an arbitrary regular language L, the set L1 can contains an in nite number of words having w as pre x even if the length of the interval Iw is zero. In this situation, all the elements of L1 having w as pre x are representing the same real number x and Iw = [x; x]. Therefore x has an in nite number of representations. To avoid this situation, we proceed as follows. We can only consider the states q such that aq > 0. We have the following rules If aq 6= 0 and there exists w such that p:w = q, then ap 6= 0. If aq = 0 and there exists w such that q:w = p, then ap = 0. Indeed, in the rst case, if p:w = q then un (p) un jwj (q). In the second case, if q:w = p then un (q) un jwj (p). Hence we obtain the conclusion by dividing both sides by Pq0 (n) n . We can split the set of states of ML into two subsets Q0 = fq j aq = 0g and Q>0 = fq j aq > 0g. If we consider only the states of Q>0 and the corresponding edges connecting those states, we obtain a new automaton accepting a new language L0 . Representations of real numbers for the numeration system built upon L or L0 are the same except that for the system built on L0 a real number has at most two representations (only when it is the endpoint of some interval Iw ). Example 19. We consider the language L accepted by the automaton depicted in Figure 5. This language is such that the number of words beginning with b (resp. a or c) has a polynomial (resp. exponential) behavior. This means that the length of the interval Ib is zero (computations are given in [11, Example 6]). Therefore the greatest word in the lexicographical ordering of L1 beginning with a represents the same real number x as any word in L1 beginning with b or the smallest word
REAL NUMBERS HAVING ULTIMATELY PERIODIC REPRESENTATIONS. . .
a
b
a Q
b
a b
a,c
0
Q
11
b
>0
Figure 5. A trim minimal automaton. beginning with c. Removing the states of Q0 , gives a new language L0 . In the numeration system built upon L0 , the number x has exactly two representations: the greatest word beginning with a and the smallest one beginning with c. In this latter system, if w is pre x of an in nite number of words in L then the length of the interval Iw is strictly positive. In the following of this paper, we shall assume that aq > 0 for all states q in ML except possibly for the sink state. 7. Determining the numbers having an ultimately periodic representation Let us have a closer look at those intervals Iw (for the details, the reader is referred to [11]). If w 2 ‘ is pre x of an in nite number of words in L then the interval Iw is given by (4) X uq :m (n ‘) X uq :m (n ‘) v(n 1) v(n 1) 0 0 + + lim ; lim n!1 n!1 v(n) v(n) v(n) v(n) ‘ ‘ m2 m<w
m2 m w
Using the fact that (see [11, Proposition 5]) limn!1 vq (n)=v(n) = aq limn!1 uq (n)=vq (n) = (
(5)
the interval Iw be rewritten as (6)
1
+
1 X
‘+1
m2 ‘ m<w
aq0 :m ;
1
+
1)= 1 X
‘+1
aq0 :m :
m2 ‘ m w
Notice that this formulation di ers slightly from [11] because we have here aq0 = 1 and the others aq ’s are strictly positive (except for the sink). Observe also that the length of Iw is ‘+11 aq0 :w > 0. Remark 20. Notice that if w 2 is pre x of an in nite number of words in L then q0 :w is a coaccessible state (so it cannot be the sink) and with our assumptions, aq0 :w > 0. Example 21. Consider the numeration system associated to the language accepted by the automaton depicted in Figure 6. Here, easy computations show that = 2, aq0 = aq2 = 1 and aq1 = 2 (to obtain the aq ’s, one has only to compute the eigenvectors of the eigenvalue of the adjacency matrix). Any word in L1 begins with a, so Ia = [1=2; 1] (instead of this reasoning, formula (6) could also be used to compute the values of the endpoints of Ia ). This interval is partitioned into three parts, Iaa = [1=2; 5=8]; Iab = [5=8; 3=4]; Iac = [3=4; 1]:
12
P. LECOMTE AND M. RIGO
c a
a 0
b
1
2
c
Figure 6. A trim minimal automaton. Thus if a real number x belongs to Ia then we have an in nite word representing x beginning with a , 2 . For the next step, we have Iaac = Iaa = [1=2; 5=8];
Iaba = Iab = [5=8; 3=4]
and Iac is split into three parts Iaca = [3=4; 5=6];
Iacb = [5=6; 7=8];
Iaca = [7=8; 1]:
Actually the form of the partition of an interval Iw into intervals Iw depends only on the state q0 :w and not on the word w itself. De nition 22. If 0
< f[
1, the strictly increasing function ; ]
: [ ; ] ! [0; 1] : x 7!
x
maps the interval [ ; ] onto [0; 1]. If z belongs to [ ; ] then we say that f[ ; ] (z) is the relative position of z inside [ ; ]. We denote by Lw (resp. Uw ) the lower (resp. upper) bound of the interval Iw . Roughly speaking, the next proposition states that two intervals corresponding to the same state are homothetic. But rst, we need a technical lemma. Lemma 23. We have, for all states q, X aq: = in particular,
aq
2
P 2
a q0 : = .
Proof. Clearly, Iw = [ 2 Iw and the length of Iw is equal to conclusion follows directly from (6).
P 2
jIw j. The
Proposition 24. Let m and w be two words such that q0 :m = q0 :w. For all 2 , the interval Im exists1 i Iw exists and the relative position of Lm (resp. Um ) inside Im is equal to the relative position of Lw (resp. Uw ) inside Iw . Proof. The interval Im exists if aq0 :m > 0. Since q0 :m = q0 :w then for all 2 , aq0 :m > 0 i aq0 :w > 0. Using (6), the relative position of Lm inside Im is given by 1 1 X 1 Lm aq0 :u = jmj+1 aq0 :m jmj+1 u2 jmj u<m
this can be rewritten as jmj
(
1) aq0 :m
( Lm
1)
X u2 jmj u<m
aq0 :u aq0 :m
1An interval I exists if there exists a word in L w 1 having w as pre x. This means that w is pre x of an in nite number of words in L.
REAL NUMBERS HAVING ULTIMATELY PERIODIC REPRESENTATIONS. . .
and using the de nition of Lm , we get 1 X 1 aq0 :u (7) aq0 :m jmj+1
X
13
aq0 :u :
u2 jmj u<m
u2 u<m
First notice that the sum over the words u of length jmj + 1 and lexicographically less than m can be split into two subsets: the words m with < and the words having a pre x of length jmj lexicographically less than m. So (7) can be written X X 1 1 X aq0 :m + aq0 :u aq0 :u aq0 :m jmj+1 jmj 2
< fA0q1 ;a : fA0q ;b : [1=4; 1=2] ! [0; 1] : x 7! 4x 1; > : f A0 1 : [1=2; 1] ! [0; 1] : x 7! 2x 1: q1 ;c The automaton FL is depicted in Figure 7. For the sake of simplicity, fA0q1 ; is denoted by f , for = a; b; c (since it does not lead to any confusion). We also put an index a or c to id to remember the corresponding letter. A path f1 t f
fc
ida Aq
fb
Aq
0
fa id c
1
Aq
2
Figure 7. The automaton FL . in FL corresponds to the composition of a ne functions ft 1 inf reversed order. Through FL , we can determine the real numbers having ultimately periodic representations. Indeed, if we consider a cycle f1 f FL starting in Aq0 then, in view of t in Algorithm 30, if the unique xed point of the corresponding function F = ft f 1 is x then fI 1 (x), with fI : x 7! x 11 and fI 1 (y) = ( 1)y+1 , has an ultimately periodic representation (we are back in the initial state q0 and since x = F (x), we have the same initial value; due to the initialization step in Algorithm 30, we have to apply fI 1 once). As an example, the xed points of fb ida , fb idc fa ida and fb fc idc fa ida are respectively 1=3, 1=15 and 5=31 and therefore 2=3, 8=15 and 18=31 have ultimately periodic representations. From the path in F L , we also know these representations: (ab)! , (aacb)! and (aaccb)! . We can also consider a cycle f1 f FL starting in Aq1 instead of Aq0 and a t in to A . Once again, let x be the xed point of F = ft path g1 from g A s q0 q1 1. f 1 From Algorithm 30, the number fI 1 g1 1 g (x) has an ultimately periodic s representation. As an example, consider F = fc idc fa fc having 3=5 has xed point. A trivial path from Aq0 to Aq1 is given by ida , so fI 1 (3=5) = 4=5 has an ultimately periodic representation: a(cacc)! . Another path in FL from Aq0 to Aq1 is (ida ; fa ; idc ), so fI 1 fa 1 (3=5) = 23=40 is represented by aac(cacc)! . Let us introduce some notation. Let q
= fw j
F (Aq ; w)
= Aq g;
REAL NUMBERS HAVING ULTIMATELY PERIODIC REPRESENTATIONS. . .
17
where F is the transition function of FL . If w = f1 f to q , we denote t belongs by Fq;w the composed function (in reversed order) ft corresponding f to w. 1 Let q = fw j F (Aq0 ; w) = Aq g; If w = f1 f to q , we denote by (Fq;w ) 1 the composition of the inverse s belongs 1 1 functions f1 corresponding f to w. s Theorem 34. Let L be a regular language satisfying our basic assumptions. Set ( 1) y + 1 : fI 1 : y 7! The set of real numbers having an ultimately periodic representation is given by ffI
1
(Fq;w )
1
(x) j 9q 2 Q; z 2
q; w
2
q
: x = Fq;z (x)g:
Proof. This is a direct consequence of Algorithm 30. Remark 35. For all states q, the languages functions are regular.
q
and
q
over a
nite alphabet of
9. Equivalence with -development Let > 1 be a Pisot number. To this number corresponds a unique positional and linear Bertrand number system U = (Un )n2N having its characteristic polynomial equal to the minimal polynomial of [2]. We denote by L the language U (N) of all the normalized representations computed by the greedy algorithm (without leading zeroes) [5]. In this section we show that this latter language L satis es the hypotheses given in Section 3. We also prove that the representations of real numbers in the abstract numeration system built upon L and the classical -developments of the numbers in [ 1 ; 1] coincide. (For a presentation of the -development, we refer the reader to [13, Chapter 7] or [14].) De nition 36. Recall that a positional numeration system U = (Un )n2N is said to be a Bertrand numeration system if 8n 2 N; w 0n 2
U (N)
,w2
U (N):
As an example, the k-ary number system is a Bertrand system. p
Example 37. The golden ration = 1+2 5 is a Pisot number, indeed its minimal polynomial is P (X) = X 2 X 1 and the other root of P has modulus less than one. The polynomial P is also the characteristic polynomial of the linear recurrence relation de ned by Un+2 = Un+1 + Un ; n 2 N: If we consider the initial conditions U0 = 1 and U1 = 2, then as a consequence of the greedy algorithm, the set of representations of the integers is U (N) = f"g [ 1f0; 01g . Due to the particular form of the language U (N), it is clear that this system (namely the Fibonacci system) is the linear Bertrand number system associated to . Consider an arbitrary Pisot number . It is well known that the -development of one is nite or ultimately periodic [17]. In the rst case, e(1) = t1 t m and we de ne, as usual, 1))! : e (1) = (t1 m t1 (tm It is clear that we still have t2 tm 1 t1 t1 + 2+ + m + m+1 + : 1= Let > 1 be a real number. The set D of -developments of numbers in [0; 1) is characterized as follows.
18
P. LECOMTE AND M. RIGO
Theorem 38. [14] Let > 1 be a real number. A sequence (xn )n 1 belongs to D if and only if for all i 2 N, the shifted sequence (xn+i )n 1 is lexicographically less than the sequence e (1) or e (1) whenever e (1) is nite. For any real number > 1, we denote by F (D ), the set of nite factors of the sequences in D . Bertrand numeration systems are characterized by the theorem of Bertrand given below. Notice that U is not necessarily linear. Theorem 39. [1] Let U = (Un )n2N be a positional numeration system. Then U is a Bertrand numeration system if and only if there exists a real number > 1 such that 0 U (N) = F (D ). In this case, if e (1) = (dn )n 1 (or e (1) = (dn )n 1 whenever e (1) is nite) then U0 = 1 and Un = d 1 Un
1
+ d 2 Un
+
2
+ d0 + 1; n nU
1:
Let > 1 be a Pisot number. First we assume that e (1) is ultimately periodic; there exist minimal integers N 0, p 1 such that e (1) = t1
(t t N +1
N
t) N +p
!
:
The Bertrand numeration system U = (Un )n2N belonging to the class of positional systems related to is a linear numeration system satisfying the recurrence relation Un
= t 1 Un
1
+(tp+1
+
p+ 1t Un p+1
t1 ) Un
p 1
+ (tp + 1) Un
+
+ (t N +p
p
tN ) Un
N p;
n
N + p:
(In other words, (Un )n2N satis es the canonical beta polynomial of [9].) In what follows, is given and we denote U simply by U . The main point is the following. Since is a Pisot number, the set F (D ) = 0 U (N) is recognizable by a nite automaton A [6] (i.e., the -shift is so c). This automaton has N + p states q1 ; : : : ; qN +p . For each i 2 f1; : : : ; N + pg, there are edges labeled by 0; 1; : : : ; ti 1 from qi to q1 , and an edge labeled ti from qi to qi+1 if i < N + p. Finally, there is an edge labeled tN +p from qN +p to qN +1 . All states are nal and q1 is the initial state. The set F (D ) is recognized by the automaton depicted in Figure 8 (the sink is not represented).
0,...,t1−1 q1
t1
0,...,tN −1
0,...,t 2−1 t2 q2
t N−1
0,...,tN+p−1 qN+p
tN
t N+p t N+p−1
0,...,t
N+1
qN
t N+1
qN+1
−1
Figure 8. Automaton recognizing F (D ) = 0
U (N).
In an abstract numeration system, allowing leading zeroes changes the representations (indeed, 0w is genealogically greater than w then valS (0w) > valS (w), see for instance [10, Example 1]). Therefore, we modify slightly the automaton A to
REAL NUMBERS HAVING ULTIMATELY PERIODIC REPRESENTATIONS. . .
19
obtain an automaton A0 recognizing exactly U (N) (i.e., without leading zeroes). To that end, we add a new state q0 . There are edges labeled by 1; : : : ; t1 1 from q0 to q1 and an edge labeled t1 from q0 to q2 . This state q0 is the initial state of A0 and is also nal. The automaton A0 is sketched in Figure 9.
q
0
1,...,t1−1 0,...,t1−1
t1 0,...,t 2−1 t1
q1
q2 A
Figure 9. The automaton A0 recognizing
U (N).
So we consider the abstract system built upon L = U (N). To be able to compute the intervals Iw related to this system with formula (4), our task is now to determine the di erent sequences uqi (n). To that end, we use the speci c form of A0 . The rst word of length n + 1 in U (N) is 1(0)n and its numerical value is Un . In the same way, 1(0)n 1 is the rst word of length n. Therefore, (9)
uq0 (n) = Un
Un
1:
0
Since q0 is the initial state of A , as usual we write u(n) and v(n) instead of uq0 (n) and vq0 (n). Since is a Pisot number and the characteristic polynomial of U is the minimal polynomial of , there exists a real number such that n
Un For n
:
1, from the form of A0 we deduce that u(n) = (t1
1) uq1 (n
1) + uq2 (n
1)
and (10)
uq1 (n) = t1 uq1 (n
1) + uq2 (n
1) = u(n) + uq1 (n
As a consequence of (10), since all the states are uq1 (n) =
n X
1):
nal uq1 (0) = u(0) = 1, we
nd
u(i) + uq1 (0) = v(n):
i=1
From (10) we also have uq2 (n 1) = uq1 (n) t1 uq1 (n 1) and thus uq2 (n) = v(n + 1) t1 v(n). But considering the path in A0 , we get uq2 (n) = t2 uq1 (n 1) + uq3 (n 1). So we nd uq3 (n) = v(n + 2) t1 v(n + 1) t2 v(n). Continuing this way, for i N + p uqi (n) = v(n + i
1)
t1 v(n + i
2)
t2 v(n + i
3)
i 1tv(n):
We are now able to determine the endpoints of the intervals Iw . It is clear from n (9) that v(n) = Un . Since v(n 1) = v(n) u(n) using (5), it is clear that
20
v(n
P. LECOMTE AND M. RIGO
1)=v(n) ! 1= if n ! 1 and therefore, for i 2 N lim
n!1
v(n i) v(n uq1 (n i) = lim n!1 v(n v(n) i + 1) v(n
v(n 1) = v(n)
i + 1) i + 2)
i
:
In the same manner, lim
n!1
Continuing this way, for j
uq2 (n i) = v(n)
1 i
:
N + p and i 2 N
uqj (n i) = n!1 v(n) lim
j i 1
t1
j i 2
j 1t
We can now compute the di erent intervals Iw . The 1; : : : ; (t1
i
t1
1); t1 ; 10; : : : ; 1t1 ; 20; : : : ; (t1
i
:
rst words in
U (N)
are
1)t1 ; t1 0; : : : ; t1 t2 ; 100 : : :
Using (4) we have the intervals corresponding to words of length one j j j j+1 uq1 (n 1) Ij = [ ; + lim ]=[ ; ]; 1 n!1 v(n)
j < t1
and
t1 t1 t1 uq2 (n 1) ; + lim ] = [ ; 1]: n!1 v(n) For the words of length two, if 1 j < t1 and 0 k < t1 then I t1 = [
k j k uq1 (n 2) j k j k+1 j ] = [ + 2; + 2 ] Ijk = [ + 2 ; + 2 + lim n!1 v(n) and
j t1 j t1 uq2 (n 2) j t1 j + 1 ] = [ + 2; ]: Ijt1 = [ + 2 ; + 2 + lim n!1 v(n) For the words of length two beginning with t1 , we have I t1 j = [ and
t1
+
j t1 j+1 ; + 2 ]; 0 2
j < t2
uq3 (n 2) t2 t1 t2 t1 t2 ; + 2 + lim ] = [ + 2 ; 1]: 2 n!1 v(n) Continuing this way, it is straightforward computation to see that we have three situations: (1) if w = w0 w qi = q0 :w0 r with r w 1 and wr < ti then I t1 t2 = [
t1
+
Iw = [
r X
wi
i 1
r X
;
i=0
wi
i 1
+
r 1
]
i=0
(2) if w = w0 wr+1 w is such that qi = q0 :w0 rw r+s r w 1 , wr < ti and wr+1 w is the maximal word read from q :w w in A0 (in other r+s 0 0 r words, q0 :w0 w q1 and wr+1 w is the pre x of length s of e (1)) r = r+s then r r+s X X wi i 1 + r 1 ]: wi i 1 ; Iw = [ i=0
i=0
(3)
nally, if w = w0
r
is w a pre x of e (1) then
Iw = [
r X i=0
wi
i 1
; 1]:
REAL NUMBERS HAVING ULTIMATELY PERIODIC REPRESENTATIONS. . .
21
Now instead of considering the abstract numeration system built upon L = (N), we can consider the classical -development of a real number x 2 [1= ; 1]. U The rst digit of e (x) is an integer j belonging to f1; 2; : : : ; t1 g. Since -developments are computed through the greedy algorithm, it is clear that the rst digit is j < t1 if and only if x 2 Ij0 = [j= ; (j + 1)= ) and it is t1 if and only if x 2 It01 = [t1 = ; 1]. So the interval Ij for the abstract numeration systems considered above and the intervals Ij0 corresponding to the greedy algorithm are the same for the rst step (except that in the abstract system, a real number can have two representations but we can avoid this ambiguity by considering intervals of the form [a; b) and therefore the two intervals Ij and Ij0 will coincide exactly). By application of the greedy algorithm, we can compute intervals Iw0 such that x belongs to Iw0 if e (x) has w as pre x. Clearly those intervals Iw0 coincide with the intervals Iw and therefore, the classical -developments are the same as the representation obtained in the framework of the abstract numeration systems (naturally, under the extra assumptions of this section corresponding to regular languages associated to Pisot number). Now we can use a result of Klaus Schmidt concerning ultimately periodic developments [17] and state the following result. Theorem 40. If L is the language of all the representations of the integers in a linear Bertrand numeration system associated to a Pisot number then the set of real numbers having an ultimately periodic representation in the abstract system built upon L is exactly Q( ) \ [1= ; 1]: Remark 41. In this section, we have only considered the case e (1) ultimately periodic. If e (1) = t1 t nite (tm 6= 0) then the same situation holds. The m is construction of the automaton A is the same as before but with N = m and p = 0. All the edges from qm lead to q1 and are labeled by 0; : : : ; tm 1. The automaton A is depicted in Figure 10.
0,...,t1−1 q1
t1
0,...,t 2−1 t2 q2
0,...,t m −1 t m−1
Figure 10. The automaton A in the case e (1)
qm nite.
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22
P. LECOMTE AND M. RIGO
[9] M. Hollander, Greedy numeration systems and regularity, Theory Comput. Syst. 31 (1998), 111{133. [10] P. B. A. Lecomte, M. Rigo, Numeration systems on a regular language, Theory Comput. Syst. 34 (2001), 27{44. [11] P. Lecomte, M. Rigo, On the representation of real number using regular languages, Theory Comput. Systems 35 (2002), 13{38. [12] L.-M. Lopez, P. Narbel, Substitutions and interval exchange transformations of rotation class, Theoret. Comput. Sci. 255 (2001), 323{344. [13] M. Lothaire, Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications, 90. Cambridge University Press, Cambridge, (2002). [14] W. Parry, On the -expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401{416. [15] M. Rigo, Numeration systems on a regular language : Arithmetic operations, recognizability and formal power series, to appear in Theoret. Comput. Sci. (2001). [16] J. Shallit, Numeration systems, Linear recurrences, and Regular sets, Information and Computation 113 (1994), 331{347. [17] K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), 269{278. [18] W. Thomas, Automata on in nite objects, in Handbook of theoretical computer science, Vol. B, J. Van Leeuwen Ed., Elsevier, Amsterdam, (1990), 133{191. [19] S. Yu, Regular languages, in Handbook of formal languages, Vol. 1, 41{110, Springer, Berlin, (1997). E-mail address: [email protected] E-mail address: [email protected] (P. L. and M. R.) Institut de Mathematique, Universite de Liege, Grande Traverse 12 (B 37), B-4000 Liege, Belgium.
