by shunji ito and shin-ichi yasutomi
TRANSCRIPT
Japan. J. Math.
Val. 16, No. 2, 1990
On continued fractions, substitutions and
characteristic sequences
[nx+y]-[(n-1)x+y]
By Shunji ITO and Shin-ichi YASUTOMI
(Received June 9, 1989)
•˜ 0. Introduction
Let x, y be positive numbers with 0<x<1 and define cn(x, y) by
cn(x, y)=[nx+y]-[(n-1)x+y] for n=0, 1, 2,....
We consider the following infinite zero one sequence:
C(x, y):={cn(x, y):n=1,2,...}
called the characteristic sequence of (x, y).
We know a characterization for homogeneous characteristic sequences
C(x, 0) given as follows.
THEOREM (Markov-Venkov [4], [6]). Let x(0<x<1) be an irrational
number and let x=[0, a1, a2,...] be its simple continued fraction expansion.
Construct sequences of finite words {Cn: n=1, 2,...}, {Dn: n=1, 2,...} as
follows:
Then the characteristic sequence C(x, 0) is written as
C(x,0)=0.Cn•EC2•EC3...,
where A...A means a string of m consecutive A's.
The purpose of this paper is to give another characterization for the characteristic sequence C(x, 0) by using a method combining substitutions and an algorithm which induces a diophantine approximation of x, and to extend this method to characterize inhomogeneous characteristic sequences C(x, y).
288 SHUNJI ITO and SHIN-ICHI YASUTOMI
In this paper, we introduce a following transformation T on X:=[0,1):
if x•¸I0=[0, 1/2),
if x •¸ I1=[1/2, 1)
and define "almost inverse transformations~0 and ¢i by
and
Then c, satisfies T o q5i=id on X and ci o T=id on Ii for i=0,1. By N we mean the set of natural numbers but 0. Definie an infinite sequence {i1, i2,
...} E {0,1}N, which is called the name of x e X, by means of the condition that
if
On the other hand, we introduce the substitutions Yi: Un=1 {0,1}n•¨Un=1 {0,1}n
for i=0,1 given by
Then we obtain the following fundamental lemma.
LEMMA. The following commutativee relation holds for each i=0,1:
for
Using this lemma, we get a characterization of C(x, 0) as the limit point of symbol 0 by iterations of substitutions, that is,
THEOREM. For each irrational number x e [0, 1), we have
Moreover we obtain a characterization of reduced quadratic numbers as fi xed points of these substitutions.
THEOREM. Let x be a quadratic irrational and reduced. Then there exists a nontrivial substitution Y such that r(C(x, 0))=C(x, 0).
This result is extended to the inhomogeneous case naturally. In fact
we introduce a multidimensional transformation U, its "almost inverses" b
(i=1, 2, 3) and substitutions rSi(i=1, 2, 3) in analogy of the homogeneous
Substitutions and characteristic sequences [nx+y]-[(n-1)x+y] 289
case, and obtain the fundamental lemma stating that
for i=1,2,3.
Using this lemma, we also get a characterization of C(x, y) as follows
where (i1,...,i) is the name of (x, y) with respect to the transformation U.
(See the section 3 in detail.)
In section 1, we discuss number theoretic properties of the transforma
tion T. This transformation seems to play very fundamental role like a
binary expansion of real numbers. At the end of section 1, we see that this
transformation gives mediants and convergents of x from below. In section
2, we introduce substitutions ro and Y; and obtain the theorem characterizing
the homogeneous characteristic sequence C(x, 0). In the section 3, we ex
tend this method to the inhomogeneous case.
•˜ 1. An algorithm and its properties
Let X=[0, 1) and let us define a transformation T on X by(1.1)
if x •¸ I0,
if x•¸I1
where I0=[0,1/2) and I1=[1/2, 1).
We define transformations and c by
(1.2) and
We see immediately that for each i e {0,1}, the map cps satisfies the following
relation:
(1.3) on and on
They are modular transformations, and we introduce the following matrices
associated with cpi:
(1.4)
290 SHUNJI ITO and SHIN-ICHI YASUTOMI
For each x E X, we define a {0,1}-sequence {in(x):n=1, 2,...} with respect to T by
(1.5) in(x)=0 if T n-1x •¸ I0,
1 if Tn-1x •¸ I1.
We call the sequence {in(x): n_??_1} the name of x with respect to T.
Let ir:X•¨{0, 1}N be a mapping defined by associating to x its name,
that is,
(1.6) ƒÎ(x)=(ii(x),i2(x),...,in(x),...),
and put the image of X by ir(X). Then it satisfies the following commutative relation:
(1.7)
where a is the shift operator on {O, 1}N defined by
for
The set of (~oi1o ® ... o cpin)(0) for all (i1, i2,...,in) E {0, 1}N is denoted by
[coo, cp1](0). The definitions (1.1) and (1.2) yield that the set [cot,, co1](0) coincides with {T-n(0):n=1, 2,...}.
PROPOSITION 1.1. Let Q be the set of rational numbers. Then we have
Q•¿[0, 1)=[_??_0,_??_1](0).
PROOF. The relation Q (1 [0, 1) D [coo, w](0) is trivial. We prove the op
posite inclusion relation by induction on the denominator m of rational
numbers. Let n/m E Q fl [0, 1) with (n, m)=1. If m=1, then cio(0)=0. If
m=2, then co,(0)=1/2. Assume that for all integers p, q such that p<q<m,
there exist i1, i2,...,ik, E {0, 1} such that p/q=(cpio o...o Wik)(0). If m>2n,
then n/(m- n) E Q (1 [0, 1) and cpa(n/(m- n))=n/m. Therefore, from the assump
tion we have n/m E [cot,, cpl](0). If m <2n, then (2n-m)/n E Q fl [0, 1] and
cpl((2n-m)/n)=n/m. Thus we have n/m E [coo, coi](0).
COROLLARY 1.1. A number x E X is irrational if and only if the cardi
narity of 1 in the name of x is infinite, i.e. #{n:in(x)=1}=•‡.
Let us define the matrices by
Substitutions and characteristic sequences [nx+y]-[(n-i)x+y] 291
(1.8) (n_??_1)
. Then, from the relation (1.3) and definition (1.4), we see
(1.9) f or
and
(1.10) for x•¸X,
that is, we have the following representation for each x E X,
(1.11) (n_??_1),
and in particular,
(1.12)
We discuss relations between convergents pn/qn of continued fraction
and above fractions tn/rn of x. Let S:X•¨X be the function defined by
where [x] is the largest integer which does not exceed x. And define func
tion pa on X for a•¸N by
Immediately we haveon
So pa is the "almost inverse" of S(x) and we identify it with a matrix a 1 1 0.
Let the continued fraction exapnsion of x be [0, a1, a2, ...] and we call ai the i-th digit of x, i.e. ai=[1/Si-1(X)1. And define the matrix representation of the comvolution pa$ o...o p by
(1.13)
292 SHUNJI ITO and SHIN-ICHI YASUTOMI
then we have the following well known formula:
We call pn/qn the n-th convergent of x, and (Apn+pn-1)/(Aqn+qn-1) mediants
of x (A=1, 2,...,an+1-1, n=1, 2,...). We know that for irrational numbers x the continued fraction expansion is unique, and for rational numbers, the
continued fraction expansion is also unique, if we restrict the length of ex
pansion to even length.In order to observe the behavior of tn/rn, we prepare the following de
composition lemma.
SUBLEMMA.
(1)
(2)
(3)
(4)
(5)
where A...A means the product of k copies of A.
The proof is easy.
LEMMA 1.1. The following decomposition holds: for a, b e N,
PROOF. This follows easily from (2), (4) and (5) in Sublemma.
Substitutions and characteristic sequences [nx+y]-[(n-1)x+y] 283
THEOREM 1.1. For each irrational x e X, let us denote the simple continued fraction of x by
x=[0, a1, a2,...].
Then the name {in(x)} nof x is given by
PROOF. By Lemma 1.1, the following decomposition holds:
that is,
Therefore, we have
Thus we have
REMARK 1.1. For each rational number x e Q fl [0, 1), from the fact that x is expanded uniquely as a simple continued fraction x=[0, a1,...,a2k] with even length, the name of rational number x is written as a word of finite length such that
Hence the assertion of Theorem 1.1 is also affirmative for rational numbers. Since the name of zero is (0, 0,...), we denote the name of x by
if we find it convenient to have the name of a rational number x as a word
of infinite length.
REMARK 1.2. It is wel known that a number x is quadratic if and only
294 SHUNJI ITO and SHIN-ICHI YASUTOMI
if the continued fraction expansion of x is periodic, and a quadratic irrational number x is reduced, that is, 0<x<1 and x<-1, where x means the algebraic conjugate of x, if and only if the continued fraction expansion of x is purely periodic (see [6]). Therefore, by Theorem 1.1, we see that the name {in(x)} of x is periodic if x is quadratic irrational, and the name is
purely periodic if x is quadratic irrational and reduced.
LEMMA 1.2. For each irrational number x E X, the pair of integers (rn, tn) in (1.12) is given explicitly as follows: let k be an integer such that
then we have
where
PROF. If n satisfies inequalities
then, from Theorem 1.1 and (1.12), (rn, tn) is given by
If n satisfies inequalities
then (rn, tn) is similarly given by
Substitutions and characteristic sequences [nx+y]-[(n-1)x+y] 295
Thus Lemma 1.2 has been proved.
Lemma 1.2 yields immediately the following theorem.
THEOREM 1.2. Let x be an irrational number of X. Then the fractions tn/rn given by (1.12) are mediants or convergents from below, that is,
PROOF. The proof is coming from the fact that the mediants or conver
gents from below is given by
and from Lemma 1.2.
REMARK 1.3. From Theorem 1.2, we may call the transformation T an
algorithm which induces diophantine approximation from below. Analogous
algorithms related to the approximation from below are found in [2] and [3].
•˜ 2. The relation between substitutions on a free group
and sequences [nx]-[(n-1)x]
Let G{0, 1} be a free group generated by two symbol 0 and 1, and we de
note by 0-1, 1-1 inverse of 0, 1, respectively.
We consider an endomorphism Y of G{0,1}, i.e., Y is determined by r(0)
and Y(1) and the relations
for
The convolution r o Y' of endomorphisms Y and r' of G{0, 1} is defined by
(r ® Y')(V)=r(r'(V)). In this section, we consider only special endomorphisms Yo and Y1 defined by
(2.1)
ƒÁ0
:0•¨0 1•¨01and
ƒÁ1:0•¨01 1•¨ 1.
These endomorphisms are indeed automorphisms because of the existence of
the inverses r-10 and Y-11:
296 SHUNJI ITO and SHIN-ICHI YASUTOMI
(2.2)
ƒÁ-10:0•¨0
1•¨0-11and ƒÁ-11:
0•¨0 1-1
1•¨1. Let W be a set of all finite words generated by {0, 1}, i.e., W= Un=1 {0, 1}n.
Then endomorphisms ri satisfy the property
(2.3) ƒÁ(W)•¼W for ƒÁ=ƒÁi.
An endomorphism r satisfying (2.3) is called a substitution. (see [5]).
Let us define a mapping C from X=[0, 1) to {0,1}N by
(2.4) C(x)=(c1(x), c2(x),...,cn(x),...),
where cn(x)=[nx]-[(n-1)x].
We call C(x) the characteristic sequence of x, named by Christoffel, and
denote the image of X by Wc.
FUNDAMENTAL LEMMA 2.1. Let r0 and r1 be substitutions defined by (2.1).
We let r0 and r1 act an infinite {0, 1} sequences naturally. Then we have thee
following commutative relation, for i=0 and 1,
(2.5) C_??_i(x)=ƒÁiC(x) for all x •¸ X.
PROOF. First discuss the case of i=0. Assume cn(x)=1. Then clearly
we have,
(n-1)x<[nx]_??_nx,
which yields
This means that
(2.6) and
where j=[nx].Assume c(x)=0. Then
[nx]_??_(n-1)x<nx<[nx]+1,
which yields
This means that
Substitutions and characteristic sequences [nx+y]-[(n-1)x+y] 297
(2.7)
where j=[nx]. (2.6) and (2.7) imply
Next, we discuss the case of i=1. Suppose cn(x)=1; then we have
(n-1)x<[nx]_??_nx,
which yields
This means that
(2.8)
where j=[nx].Lastly suppose cn(x)=O. Then we have
[nx]_??_(n-1)x<nx<[nx]+1,
which yields
This means that
(2.9) and
where j=[nx]. (2.8) and (2.9) imply
THEOREM 2.1. The following diagrams commute for each i=0 or 1,
(1)
and
(2)
298 SHUNJI ITO and SHIN-ICHI YASUTOMI
where W(i) is the image of Ii by C.
PROOF. The relation (I) is nothing but Lemma 2.1. Mappings rci:X•¨Ii,
C: X•¨W, C: Ii•¨Wc(i) and ri: Wc•¨Wc(i) are bijective, and satisfy the
commutative relation (1). Therefore Theorem 2.1 is now clear.
From the definition (2.1), for any sequence (i1, i2,...in,...)e{0,1}N the
word (ri, o...aYin+1)(0) is decomposed, identifying words with elements in
G{0, 1} as follows:
where
if in+i=0,
if in+1=1.
Therefore, for each w=(i1, i2,...) the limit
is well defined.
THEOREM 2.2. (1) For each irrational number x e X, we have
where {ij} is the name of x.
(2) For each rational number x e X,
where index (i1,...,in) is as indicated in Remark 1.1.
PROOF. We know in Corollary 1.1 that the name of each irrational
number satisfies the property #{n: in=1}=•‡. From Theorem 2.1, we have
for all n_??_1.
Since C(x)=(0,...) holds for all x e X by definition (2.4), C(x) is characterized as a limit of r o r~20...rin(0) for each irrational. In the case of rational
numbers, Remark 1.1 gurantes the existence of an integer n such that
This is the same as saying that
Substitutions and characteristic sequences [nx+y]-[(n-1)x+y] 299
Thus Theorem 2.2 has been proved.
For any pair of positive integers (m, n), we define an endomorphism rm,n of G{0, 1} by
hence we have
By Theorem 1.1 and 2.2, the characteristic sequence C(x) of x is given by
(2.10)
where ai (i_??_1) is the i-th digit of the simple continued fraction expansion of x.
Put
(2.11)
Then Ta1,a2 is written by
Define words 0, (j_??_3) as follows for n_??_1:
(2.12)
Then, by (2.11) and (2.12), we have the following,
(2.13)
300 SHUNJI ITO and SHIN-ICHI YASUTOMI
Therefore, by (2.10) and (2.13) we have
that is
COROLLARY 2.3. Let 8n be words defined by (2.11), (2.12). Then for each irrational number x in X, the characteristic sequence C(x) is given by
REMARK 2.1. Markov remaked in [4] the following representation for the characteristic sequence C(x). Let as be a substitution such that
then C(x) is decomposed in G{0, 1} as
where ai are digits of the simple continued fraction of x.This is so the called Markov word property. Therefore, our corollary
gives another decomposition of C(x).
THEOREM 2.4. (1) Assume that a number x e X is quadratic irrational and reduced as in Remark 1.2. Let K be the length of the period of {in(x): n_??_1} which is the name ir(x) of x, then we have
That is, characteristic sequence C(x) is a fixed point for the substitution r=
ri1 p ..._??_rjk.
(2) Assume that a number x e X is quadratic irrational, and let the name
{in(x): n_??_1} of x be (i1, i2,...,iN,...,iN+K-1) where we assume that Nand K are minimal. Then we have a relation
That is, characteristic sequence C(x) is a fixed point of the automorphism
PROOF. From the assumption of (1), the number x satisfies the relation:
Substitutions and characteristic sequences [nx+y]-[(n-1)x+y] 301
(2.14)
Therefore, by Theorem 2.1 (1), we obtain the conclusion. If the continued fraction expansion of a quadratic number x is not purely periodic, then from Theorem 1.1 y:=TN-1 x is purely periodic with peirod K. By virtue of (1.3),
(2.14), this is equivalent to
and
Thus we have completed the proof.
•˜ 3. Characteristic sequences of inhomogeneous liner forms
In this section, the characterization of sequences C(x, y):={cn(x, y)=
[nx+y]-[(n-1)x+y]:n=1, 2,...} is discussed as an analogue of the homo
geneous case. For this purpose we introduce the transformation U and sub
stitutions i (i=1, 2, 3) as follows:
Let D be
(3.1) D={(x, y)|x, y_??_0, x+y_??_1]/{(1, 0), (0, 1)}.
And let D (i=1, 2,3) be
(3.2)
and
then U1= ,2,3 Di=D and {the interior of D1} (1 {the interior of Dj=0 (i•‚j).
Figure 3.1
302 SHUNJI ITO and SHIN-ICHI YASUTOMI
Let us define a transformation U on D by
if (x, y) e D1,
(3.3) if (x, y) e D2,
if (x, y) e D3,
then the transformations U|Di: Di•¨D are bijective.
Define ~'2 and (b3 by
(3.4)
Then we have
(3.5) U_??_ƒ³i=id on D and ƒ³i_??_U=id on Di.
Let us define the name {i(x, y)|n_??_1} of (x, y) with respect to the transformation U by
(3.6) in(x, y)=i if Un-1(x, y)•¸Di.
REMARK 3.1. We observe the behavior of transformation U on bound
aries. Let us denote the pieces of boundaries as follows:
Then we see that
(1) U(„C(1,3))=B1, U(„C(2,3))=B2.
(2) U(B0)=B0, U(B1)=B0•¾B1.
(3) For any (x, y) e B1, there exists n=n(x, y) such that Ui(x, y) •¸ B1
1_??_i_??_n-1 and Un(x, y) e B0.
(4) Restriction of U on BD coincides with the transformation given by
(1.1).
Substitutions and characteristic sequences [nx+y1-[(n-1)x+y) 303
Therefore, if there exists n such that Un(x, y) is in l'(1 ,3) or P(2,3), then there exists n1 such that for m_??_n1 Um(x, y) is in Bo U B1. That is, if in(x, y) has two values, im(x, y) has only one value for large m.
We introduce the substitutions on G{0, 1} by
(3.7)
Note that V1=Yo and 52= rl in section 2. Let us define the characteristic se
quence C(x, y) of (x, y) by
cn(x, y):=[nx+y]-[(n-1)x+y],
and
C(x, y):=(c1(x, y),c2(x, y),..., cn(x, y),...).
From the definition, cn(x, y) is 0 or 1, and C(x, y) is an element of {0,1}N.
Wedenote the image of W by the mapping C by Wc. Then we have the
following lemma corresponding to Lemma 2.1.
LEMMA 3.1. Let oi(i=1, 2, 3) be substitutions defined by (3.7) and we let
5i (i=1, 2, 3) act on {0,1}N naturally. Then we have the following relation for
each i=1, 2 and 3:
(3.8) C(ƒ³i(x, y))=ƒÂiC(x, y).
PROOF. We consider the case of i=1. Let us assume c(x, y)=1 which
is equivalent to
(n-1)x+y<[nx+y]<nx+y.
Then we have the following inequalities:
which imply by definition,
and
where j=[nx+y].Next, let us asume that cn(x, y)=0 which is equivalent to
304 SHUNJI ITO and SHIN-ICHI YASUTOMI
It yields
cn+j(ƒ³1(x, y))=0,
where j=[nx+y].
Thus, we have the commutative relation:
Cƒ³1(x, y)=ƒÂ1C(x, y).
Other cases (i=2 and 3) are obtained analogously.
THEOREM 3.1. The following diagrams are commutative for each i=
1, 2, 3
(1)
and
(2)
where the inverses o-1i(i=1, 2, 3) are given by
and
PROOF. The proof is obtained analogously to the proof of Theorem 2.1.
THEOREM 3.2. Let us write Un(x, y)=(xn, yn) and denote the name of (x, y)
by (i1, i2,...).
(1) If xn•‚0 for all n, then the characteristic sequence C(x, y) is given by
the limit:
(2) If there exists an integer n such that xn=0, then the characteristic sequence C(x, y) is purely periodic and its period is given by
Substitutions and characteristic sequences [nx+y]-[(n-1)x+y] 305
PROOF. As discussed in section 2, there exists the limit (o o o ~~n}(0)
if and only if #{n|in=2 or 3}=•‡. And this is equivalent to the fact that
xn•‚0 for all n, that is Un(x, y)_??_B2. On the other hand, by Theorem 3.1, we
have
(3.9)
and C(xn, yn) begins with 0 for all n. Therefore we have the conclusion.
REMARK 3.2. In the paper [1], we noted that there exists an integer n such that xn=0 if and only if x is a rational number.
REMARK 3.3. We associate each (P (i=1, 2, 3) with the matrix Ai as follows:
and let us define matrices associated with the name {in: n_??_1} of (x, y) by
Then we obtain the following formula:
Therefore, the transformation U may be regarded as one of the algorithms which induce multi-dimensional approximation in analogy to the continued fraction algorithm. The following result is obtained for transformation U in [1]: The name of (x, y) is periodic if and only if x is quadratic and y e
Q(x) where Q(x) is the quadratic field generated by x. Therefore, we have the following theorem for the inhomogeneous case corresponding to Theorem 2.3 for the homogeneous ones.
THEOREM 3.3. Assume that a real number x is quadratic and y e Q(x),
306 SHUNJI ITO and SHIN-ICHI YASUTOMI
then the name of (x, y) is periodic, and thcrefore the characteristic sequence C(x, y) is a fixed point of an automorphism:
where k is the length of the period and N is the initial index of the period .
References
[1] Y. Hara and Sh. Ito, A characterization of real quadratic fields by a diophantine algorithm, (preprint).
[2] Sh. Ito, Algorithm with mediant convergents and metrical theory, Osaka J. Math., 26 (1989), 557-578.
[3] Sh. Ito, On a diophantine approximation of real numbers from below, Proc. Prospects of Math. Sci. World Sic. Pub., (1988), 41-62.
[4] A. A. Markov, Sur une question de Jean Bernoulli, Math. Ann., 19 (1882), 27-36.[5] P. Michel, Coincidence values and spectra of substitutions, Z. Wahrscheinlich
- keitstheorie und Verw. Gebiete, 4,2 (1978), 205-227.
[6] B. A. Venkov, Elementary number theory, Woltevs-Noordhoff publishing co, groningens, the Netherlands, 1970.
SHUNJI ITODEPARTMENT OF MATHEMATICS
FACULTY OF SCIENCETSUDA COLLEGE
TSUDA KODAIRA 187
SHIN-ICHI YASUTOMINTT SOFT Co.
YAMASITA YOKOHAMA 231