By Shunji ITO and Shin-ichi YASUTOMI

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)


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)


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.


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


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


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:


(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


(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)


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


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)


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:


(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


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


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


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),


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

By Shunji ITO and Shin-ichi YASUTOMI

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