Combinatorial properties of infinite words associated with cut-and-project sequences

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

L

OUIS

-S

ÉBASTIEN

G

UIMOND

Z

UZANA

M

ASÁKOVÁ

E

DITA

P

ELANTOVÁ

Combinatorial properties of infinite words associated

with cut-and-project sequences

Journal de Théorie des Nombres de Bordeaux, tome 15, n

o

3 (2003),

p. 697-725.

<http://www.numdam.org/item?id=JTNB_2003__15_3_697_0>

© Université Bordeaux 1, 2003, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Combinatorial

_properties

of

infinite words

associated

with

_{cut-and-project}

_sequences

par

LOUIS-SÉBASTIEN

GUIMOND,

ZUZANA

MASÁKOVÁ

et EDITA

PELANTOVÁ

RÉSUMÉ. Le but de cet article est d’étudier certaines

_propriétés

combinatoires des suites binaires et ternaires obtenues _par

_"coupe

et

_projection".

Nous considérons ici le _processus de coupe et

pro-jection

en dimension deux où les _sous-espaces de

_projection

sont en

position

générale.

Nous _{prouvons que} les distances entre deux

ter-mes

_adjacents

dans une suite ainsi obtenue _prennent

toujours

soit

deux soit trois valeurs. Une suite obtenue par coupe et

_projection

détermine ainsi de manière naturelle une suite

symbolique

(mot

infini)

sur deux ou trois lettres. En fait ces suites _peuvent aussi être obtenues comme

_{codages d’échanges}

de deux ou trois inter-valles. Du

_point

de vue de la

complexité,

la construction par coupe et

_projection

donne des mots de

_{complexité n}

+

1, n

+ constante et 2n + 1. Les mots binaires ont une

complexité égale

à n + 1 et sont donc sturmiens. Les mots ternaires ont une

_complexité

égale

à n + constante ou 2n + 1. Une coupe et

_projection

a

trois

_paramètres

dont deux

_spécifient

les _sous-espaces de

projec-tion,

le troisième déterminant la bande de _coupe. Nous classifions les

_{triplets qui correspondent à}

des mots infinis combinatoirement

équivalents.

ABSTRACT. The aim of this article

_is

to

_study

certain combinato-rial

_properties

of infinite

_binary

and _ternary words associated to

cut-and-project

sequences. We consider here the

cut-and-project

scheme in two dimensions with

_general

orientation of the

project-ing subspaces.

We prove that a

cut-and-project

_sequence

_arising

in such a

_setting

has

_always

either two or three _types of distances between

_adjacent

_points.

A

_{cut-and-project}

_sequence thus

deter-mines in a natural _way a

symbolic

_sequence

(infinite word)

in two

or three letters. In

fact,

these _sequences can be constructed also

_by

a

_coding

of a 2- or 3-interval

_exchange

transformation.

_According

to the

_complexity

the

_{cut-and-project}

construction includes words with

_{complexity n}

+

_{1, n}

+ const. and 2n + 1. The words on two

letter

_alphabet

have

_complexity

n+1 and thus are Sturmian. The

ternary words associated to the

_{cut-and-project}

_sequences have

complexity n

+ const. or 2n + 1. A

cut-and-project

scheme has three _parameters, two of them

_specifying

the

_{projection subspaces,}

the third one

determining

the

_{cutting strip.}

We

classify

the

triples

that

_correspond

to

_{combinatorially equivalent}

infinite words.

1. Introduction

In this _paper we

study

certain

_ternary

generalizations

of

binary

Stur-mian sequences

[14,

15].

Sturmian sequences have several

equivalent

char-acterizations. One of them is

_using

the so-called

_{cut-and-project}

scheme.

Consider a lattice

Z2 ,

a

straight

line _{p : y} = _ax with _an irrational

slope

a, and a

_{strip parallel}

to the line _p of such a width that the intersection

of the

_strip

with the x-axis is a

line-segment

of

length

1. The

orthogonal

projection

of lattice

_points

in the

_strip

on the line _p determines a

_point

set in R with

_only

two

_possible

distances between

_neighbours.

If we

_represent

these distances

_by

letters A and B

_respectively

and _arrange these letters

according

to the order of distances in the

_point

_set,

we obtain a Sturmian

infinite word.

Replacing

the

_orthogonal

_projection

in the above scheme

_{by projection}

in _any direction _may cause the

_change

in

_ordering

of

_projected

_points

on

the

_straight

line _p and thus the distances between

_neighbours

may

possibly

take more than two values. Our

_{generalization}

of Sturmian words stems in

allowing

any width of the

strip

and any

projection

in the

cut-and-project

scheme.

_Sequences

of

_points

obtained in the described way are called

cut-and-project

sequences.

In our _paper we use the notion of

_geometrical

_similarity.

Two

_point

sets

are said to be

_{geometrically similar,}

if one is an affine

_image

of the other.

Geometrical

_similarity

is an

_equivalence

relation. One of the main results

of the

_present

_paper is the characterization of classes of this

_equivalence,

cf. Theorem 6.3.

’

Our construction of

_{cut-and-project}

_sequences is a

_special

case of the

definition of the so-called model sets.

_Properties

of

_general

model sets have been

_extensively

studied because

_they

are suitable models for materials

with

_long-range

_aperiodic

order. For

_example,

it is well known

_[18]

that every model set has a finite number of local

configurations.

In

particular,

for our one-dimensional case it means that there are

only

finitely

_many

different distances between

_{adjacent points.}

In our article we show that the distances between

neighbours

in a

cut-and-project

sequence take at least two and at most three values. Similar

problem

was studied

_{by Langevin}

in

_[13]

as a

generalization

of the well

known three-distance theorem

_[21].

The theorem is a

_simple

_consequence

To a

cut-and-project

_sequence we can associate a

symbolic

_{sequence in} a two- or three-letter

alphabet. Obviously,

geometrically

similar

cut-and-project

sequences determine identical

symbolic

sequences. The

opposite

statement is however not true. Even two

_{cut-and-project}

_sequences that

are not

_{geometrically}

similar _may

_provide

_coinciding

words. It is therefore

practical

to introduce the notion of combinatorial

equivalence.

In

Theo-rem 7.5 we describe the

equivalence

classes.

Another result of this paper is the

description

of

complexity

of words associated to

_{cut-and-project}

sequences

(Theorem 5.3).

We show that the

binary

cut-and-project

words have

_complexity

n + 1 and therefore are

Stur-mian ;

the

_ternary

words

_correspond

to

_codings

of three-interval

_exchange

transformations with

_permutation

_{(3, 2,1).}

2. Words

A finite

alphabet

is a set of

_symbols

.A = A concatenation

w of letters is called a word. The

length

of a word z,v is the number of letters

from which w is formed. We denote

_by

,A.* the set of words in the

_alphabet

,~4. A _one-way infinite word u is a _{sequence u} =

(Un)nEN =

_ulu2u3 ... ·

with values in .A. One _may consider also bidirectional infinite words as

sequences u =

(un)nEZ-

In our article we work with bidirectional infinite

words. Two words in an

alphabet

,A and

(vn)nEz

in an

alphabet

x3 are

_{combinatorially equivalent}

if there exists a

bijection

h : an _{no E Z such that}

h(un)

=

vn+no for all n E Z.

Let i E

_Z,

n E N. A concatenation _{uiuz+1... uZ+n_1} is called a factor of u of

_length

n. We define the

_density

of a factor w of u of

length n

as

if the limit exists.

The function that

_assigns

to a

_{positive integer n}

the number of different

factors of

_length

n in an infinite word u is called the

complexity

of _u,

usually

denoted

_{by C,}

For the

_complexity

of an infinite word in a finite

alphabet

.,4 one has

The

_complexity

function is thus a measure of disorder in the infinite word.

For more about the

_complexity

function see

_[3,

_7,

10].

Let us recall some facts about _one-way infinite words. We _say that an

infinite word U = _ulu2u3 ... in a finite

alphabet

A is

eventually periodic,

if

there exist finite words _wo, wl in ,~1.* such that w =

WOWIWIWI .... A word

which is not

_{eventually periodic}

is called

_aperiodic.

It is well known that a

such that

_G(n)

n,

(see [17, 9]).

It follows that the most

simple aperiodic

one-way infinite words are those of

complexity

C(n)

= _{n +} 1. Such words

are called Sturmian. A nice _survey of Sturmian and related _{sequences may}

be found in

_[17,

_15,

_14].

_{Algebraically,}

every Sturmian word can be written

as a _sequence of 0’s and l’s

_{given by}

one of the

_following

formulas

for

_of,/3

E

_~0,1),

a irrational. Note that the above

expressions

can be

extended for n E Z so that

_they

define

_aperiodic

bidirectional infinite words.

J.

_Cassaigne

defines in

_[8]

the

_{quasisturmian}

sequences as such infinite

sequences for which there exist

integers

k and no such that the

complexity

function is

_C(n)

= _{n +} k for _n &#x3E; no. An

example

of _a

quasisturmian

sequence is a _sequence that arises from a Sturmian word if one substitutes

every 0

by

a finite word _wo and _{every 1}

by

a finite word _wl.

Cassaigne

has

shown that _every

_{quasisturmian}

_sequence

_is,

_up to a finite

prefix,

of this

kind.

Using

the

_powerful

notion of the

_Rauzy

_graphs,

the words with

complex-ity

2n+ 1 can be divided into four classes

_according

to the maximal

_indegree

and

_outdegree

in the associated _sequence of directed

_graphs.

Arnoux and

Rauzy

gave

geometrical

characterization of infinite words of

type

3-3. We show that

_{cut-and-project}

sequences are

geometrical

reprezentations

of

in-finite words of

_type

2-2.

In order to

_explain

the classification of _sequences with

_complexity

2n + 1

into four groups, let us recall the definition of a

_Rauzy

_graph.

A

_Rauzy

graph

for a

_given

infinite word u and

_{given integer n}

is an oriented

_graph

rn

=

(Vn, En),

where the vertices are

_{represented by}

factors of

_length

n

and the oriented

_edges

are determined

by

factors of

length

n+1. An

_edge

e

starts in a _{vertex vi} and terminates in a vertex v2 if the factor vl of

length

~ is a

prefix

of the factor e and _v2 is its suffix. Thus

#Vn

=

C(n)

and

#En

=

C(n

₊

1).

The number of

edges

that start

(end)

in a vertex v is called the

_outdegree

_(indegree)

of v. For a bidirectional infinite

word,

_every

vertex of _every

_graph

has both in- and

_outdegree

at least one. For words

of

_complexity

2n + 1 in every

graph

rn

the number of

edges

is

equal

to the number of vertices

_plus

2. Therefore the maximal

_outdegree

for the

_graph

is either 2 or 3. The same statement is true for maximal

_indegree.

The

pair

maximal

_{indegree -}

maximal

_outdegree

for words of

_complexity

2n + 1

can therefore have

_only

four values. This classification holds for sequences for which the

_indegrees

are

_always

non-zero. This is the case if the word is

bidirectional or _one-way recurrent. Arnoux and

_Rauzy

in

_[4]

show that if for some _no the maximal

_outdegree

in the

_graph

is

_2,

then it is 2 for every

graph

&#x3E; no. The same statement

_again

holds for

_indegree.

In

the article the authors find

_geometrical

_{representation}

of words with

Rauzy

graphs

of the

_type

3-3 for every n E N. The infinite words considered in

this article are of the

_type

2-2,

which means that

_starting

from a certain

no both

maximal indegree

and maximal

outdegree

are

equal

to 2.

3.

_{Cut-and-project}

sequences

Generally, cut-and-project

sets are defined as

projections

of lattices of

arbitrary

dimensions

_[18].

In this _paper we

study

the

simpliest

case. The

cut-and-project

scheme in

1I~2

is

_{given by}

two one-dimensional

subspaces

Vi

and

_I§

and

_{by projections}

7r,: and 7r2:

}R2 ~ V2

which

satisfy:

1)

xi restricted to

Z~

is an

_injection.

2)

1I"2(Z2)

is dense in

_V2.

The scheme is illustrated

_by

the

_following

_picture.

In this scheme

_1rl(Z2)

and

_1r2(Z2)

are additive abelian groups. The

bi-jection

between

_them,

_{~ri 10 ~r2}

is

_usually

denoted

_by

* and called the star

map. Its inverse is denoted

by

-*.

Let

_Ví

be the linear _span of a vector

_~1

=

(1, é)

and

V2

the linear

span of a

vector x2

=

(-1, ~7).

In order _to

satisfy

conditions

1)

and

2)

_we choose

,-7 17 irrational -7/.

Any

vector

(p, q)

E

7G2

can be written as

For

_simplicity

we omit the common factor

1/(e + 7~) (which

corresponds

to different normalization of

_{vectors -}

and

_~2)

and consider the abelian groups

’

The star _map *:

Z[77]

-

_7G(e~,

_{given by}

is an

_isomorphism

of the two _groups.

_Using

this formalism we can

easily

define a

_{cut-and-project}

_sequence as

where f2 is a bounded

_interval,

called the

_acceptance

window. We denote the

_length

of an interval f2

_by

101.

An

_example

of the construction of a

FIGURE 1. Construction of a

_{cut-and-project}

_sequence.

It is well known

_[18]

that _any

_{cut-and-project}

set is Delone with finite number of

_polygons

in Voronoi

_tiling

of _space. For C R it

im-plies

that there exists an

_increasing

_sequence

(Xn)nEZ

such that

E~~~(SZ) _

E

_Z}

and the set of tiles T =

Itn

= _E

Z}

is finite. If we

_assign

to each tile t E T a letter

_h(t)

from an

alphabet

A,

the

sequence can thus be viewed as a bidirectional infinite word in

the finite

_alphabet

A.

For the

_description

of we can use a different

increasing

_sequence

xn

:= _xn+p for

arbitrary

fixed _p E Z. The

_{corresponding}

_sequence has then the

_property

=

tn+p

for all n E Z. The word

_(h(in))nEZ

is thus a

p-shift

of the infinite word

_(h(tn))nEZ.

In order to avoid

_{specification}

of the

point

indexed

_by

_0,

we introduce the

symbol

for the entire class of

all bidirectional infinite

_words,

that are associated to

_{E~~~ (SZ).}

In

_[17]

the infinite bidirectional words without

_{specific origin}

are called

_{trajectories.}

The

_terminology

_{pointed/non-pointed}

words is also used.

We are interested in combinatorial

_properties

of words In

par-ticular,

we

_study

- the

cardinality

of the

_alphabet

E

_T~

C

_,A.,

in case when

differ-ent tiles are

_assigned

to different

_{letters, i.e.,}

the

_{map h}

is

_injective,

- the subword

complexity

of

-

the

_triples

of

_parameters

_ë,

_{11, !1}

that

_give

the same bidirectional

infinite

_words,

_up to the choice of the letter

_{assignment h}

into the

alphabet

A.

4. Distances in

In this section we first determine the number of different tiles in

and then we derive the

lengths

of these tiles.

Let us first focus on

_{cut-and-project}

_sequences with

_symmetric

accep-tance intervaj f2 =

(-d, d),

with d _&#x3E; 0. Let 0 _{x 1 X2 ...}

de-note the

_positive

elements of the set

_E,,,7(-d,

_d).

Let _xk be the minimal

positive

element of

_E,,,7(-d,

_d)

with

_sign

_{xk =1=}

_sign

_xi.

Since the im-ages

x*

cover

densely

the interval

(-d, d),

such an _Xk must exist. Then

0 xZ - Xl xi xk for all i =

2, 3, ... ,

k - 1 and

(zj - xl)*

_E

(-d, d).

Thus Xi -

xl are

positive

elements of

E(-d,

d)

which is

possible only

as

xi - x1 _{= Xj} for

some j

=

1, 2, ... ,

i - 1. This

implies

that

Moreover,

Indeed,

suppose that

xk - x1 - (zk -

(-d, d),

Le.,

xk - zi E

Eë,1J( -d, d).

Since Xk &#x3E; ~ 2013 xi &#x3E;

0,

according

to

(2)

there must exist a

j

E

_{{0,1,..., A? - 1},}

such that _{Xk -} xi =

jxi,

which

gives

xk = ( j

₊ The latter is a contradiction with the

_assumption

sign

x¡ =1=

sign

_{xi .}

Theorem 4.1. For _any irrational numbers

_{~, r~, ~ ~}

_2013~~ and _any bounded interval 11 =

[c, c-I-d),

there exist

_positive

numbers

Ai

and

A2

such that the distances between consecutive

_points

in take at most three values among

A2,

Ai

+

Proof.

The

_points

of +

_d)

form an

increasing

_sequence

i.e.,

+

_{d) -}

_I

n E

_Z}.

A

_positive

number A is a distance

between consecutive

_points

of c +

_d)

if there exists such that

0 = For

any tile A &#x3E; 0 we have l:1 E

_{E~,~(-d, d).}

_Moreover,

let

both 6 &#x3E; 0 and

_16,

i E

_N,

i

_{&#x3E;_}

_2,

_belong

to

_{E-d, d).}

Then 16 is not a tile

in

_Eë,1J[C,

c +

_d),

since if x and x + 16 are elements of c +

_d)

then also

z + 6

_belongs

to and hence

z + 16

is not the closest

_neighbour

of x.

’

Denote

_{by Ai}

the smallest

_positive

element of

_{~E~~(-d, d)}

with

_ði

&#x3E;

_0,

~2

0.

_Formally,

The values

_ill

and

_A2

are the two smallest candidates to be

_lengths

of tiles in c +

_d).

According

to

₍₃₎

we have

Let _{xl E}

_{E~,,~ ~c,}

c +

_d)

such that

_xi

E

_[c,

c + d -

_{AT). 1}

Since

_xi

+

_Di

E

[c,

c +

_{d) and xl}

+

_A*

c + d -

_A*

+

_A*

c

(compare (4)),

the

_point

xj +

_O1

_belongs

to c +

_d)

and xl +

_Ai

is the closest

_{right neighbour}

of xl.

’

Let X2 E such that

_xi

E

_Similarly,

E

c, c

+

_{d) and x*}

+

_{[c, c}

+

_d),

therefore x2 +

A2

is the closest

_right

neighbour

Of

X2-Now let X3 E

Ec,

c +

_d)

such that

_x3

E

_[c

+ d - c - Then

and the

_point

X3 +

_Ai

+

_A2

_belongs

to c +

_d).

We want to show that

x3 +

O1

+

A2

is the closest

_{right neighbour}

of the

_point

x3.

Suppose

that there exists a tile

A,

such that

Combining

the

_inequalities

we obtain

Consider the

_{point 6}

:=

Ai

+

A2 - A

&#x3E; 0. Since A &#x3E;

_{Ai, A2}

_(as

_~1,

_A2

are the two smallest candidates for

_lengths

of

_tiles),

it holds that 0 6

.nIA1,

A2}

but

_using

₍₅₎

we

_get

It means that J E

_F,,,,7(-d,

_d),

which contradicts the definition of

_~1

and

A2

as

_being

the smallest elements of

EE~~ (-d,

d).

As a

_result,

the closest

right neighbour

of X3 E

ê,[C,

c +

_{d), with x*}

E

_[c

+ d - c -

2)’

is the

point

x3 +

A1

+

02.

By

that we have determined the

_right

neighbours

of all the elements of

~ê,r¡[C,

c +

_d)

and the

_proof

is

_completed.

0 Let us make several comments on the above theorem:

~ From the

proof

of the above theorem it follows that

~1

and

A2

de-pend only

on the

length

d of the interval 0 =

[c,

c +

_d)

and not on

the

_position

of the

_point

c. The

lengths

of tiles

~1

and

A2

satisfy

A]A]

0 and we shall

always

denote

by

~1

the tile with

positive

star

_image

_A]

&#x3E;

_{0, similarly,}

_~2

0.

~ If the

acceptance

window were an interval f2 =

(c,

c +

_d~

or Q =

(c,

c +

_d)

the

_proof

of Theorem 4.1 could be

_{repeated identicaly;}

for the case S2 =

[c, c+d]

a minor amendment is needed. Hence for _every

acceptance

interval 12 with

_non-empty

interior the set

_E,,,7(f2)

has at

most three

_types

of tiles.

e Theorem 4.1

implies

that _every

_segment

of the set has three

types

of distances

_{~1, A2,}

~1

+

_A2

between

_neighbouring

_points.

More

_precisely,

the set

has at most three distances between

_neighbours

for

_arbitrary

irra-tional _c,

_{t7, - 0}

-77, and

arbitrary

bounded intervals

J,

f2. For a

given

positive

irrational a and a

positive integer n

we

_{define 17}

= _-a,

ê

= -~,

J =

(0,1~,

and Q =

~1, K),

where K =

[a(n

₊

1)]

₊

a(n+1)+1.

Then

The condition 0 a - ba 1

_implies

a = 1 +

(ba),

hence _{a -} ba =

1 -

_{ba}.

We also have

We thus obtain M =

{1 -

1~

_ ~0,1, ... , n}.

The _set M has

at most three distances between

_neighbours.

The same

_property

is

satisfied

_by

1- M =

1~

=

_{0,1, ... ,}

n},

which is the well known three-distance theorem

_[21].

From the

_proof

of Theorem 4.1 we _may derive a

prescription

to find the

_{right neighbour}

of a

_given

_{point x}

in the

_{cut-and-project}

_sequence,

according

to the

_position

of x* in the

_acceptance

interval.

Corollary

4.2. Let {1 =

[c,c

_-I-

d)

be _a bounded

interval,

and let

_A2,

and

~1

+

_A2

be the tiles in such that

_Lli

&#x3E; 0 &#x3E;

_02.

The closest

In the

_remaining

_part

of this section we derive the

lengths

of tiles in

for a

_given

interval {1 =

[c,c

+

_d)

and

_positive

e, q. As it will be

seen from the

_study

of

_geometrical

similarities of

cut-and-project

_sets,

the

assumption

of c, 77

positive

does not cause a loss of

generality.

Changing

continuously

the

_length

d of the

_acceptance

interval

S2 =

[c,c

₊

d)

_causes discrete

changes

of the

triplet

of distances

A2,

01

+

_A2).

Let c +

_d)

be a

generic cut-and-project

_sequence, and let

A*

&#x3E;

_0,

_OZ

0 and

_A*

+

₀₂

be the _{star map}

_images

of its tiles.

Denote

_by

_d+

the smallest number such that d

_d+

and at least one of

the distances

_{Ai ,}

A2,

O1

+

A2

does not occur in c +

_d+).

_Similarly,

denote

by

d- the

_largest

number such that d- d and at least one of

the distances

_{Ai , A2,}

_O1

+

_A2

does not occur in c +

_d-).

Due to

Corollary

4.2,

we have

d+

=

A* - 02

and thus

_E~~~(c,

c +

_d+)

has

_only

two

distances,

namely

Ai

and

_A2.

Hence

_growing

the

_length

of the

_acceptance

interval,

the

_largest

distance

_O1

+

_A2

_disappears.

Similar situation occurs in c +

_d-).

One of the distances

_ill,

A2,

O1

+

_A2

_disappears,

but it

_happens

in such a _way that the

_remaining

distances have their star map

images

of

opposite sign.

Thus

[c,

c +

_{d_ )}

has distances

From

_Corollary

4.2 we obtain

respectively.

Starting

from a

given

initial value

do,

for which the set c +

_do)

has 2

_tiles,

we can determine

_by

recurrence all

_lengths

dn, n

E

_Z,

of the

acceptance

windows for which c +

_dn)

has

_only

two tiles. The initial value

_do

is determined below in

_Example

1.

Let &#x3E; 0 and 0 be the star

_images

of distances

_occurring

in the _sequence

_dn),

_{i.e., dn}

=

~~2.

Similarly,

the

_algorithm

which from the

_triple

_dn,

An2

finds the

_triple

7 A(n-1)2

has the inverse form

We have

_yet

to determine some initial values for the recurrences

(6)

and

_(7).

It turns out that it is reasonable to start with a

cut-and-project

sequence whose

acceptance

interval is of unit

_length.

Since the distances do not

_depend

on the

position

of the

interval,

we can focus on the

_acceptance

interval

_{[0, 1).}

Example

1. Let _-,17 be fixed

_positive

irrational numbers. Let us

_study

the _sequence

_{E~[0,1) =}

_{p

_p,

_q E

_{Z, 0 p -}

qé

1}.

Elements of this set have to

_satisfy

_{qé :5 p}

1 + _qe. Therefore we can write

From the above formula one _may

easily

observe that the tiles in a

~ê,f1[O, 1)

form a Sturmian word with

_slope

ê.

_Indeed,

since is an

_increasing

sequence, the

lengths

of tiles in

E,,,7[0, 1)

can be

computed

as

The sequence

_E~~~[0,1)

has thus two distances ordered as a Sturmian word

with

_slope

a = _e and shift

_{intercept /3}

=

0,

cf. definition of Sturmian words

_(1).

We denote the

_length

of the

_acceptance

interval

_by

_do

= 1. In the notation of the recurrence relations

(6)

there is

It is now obvious that for the determination of the

_lengths

of tiles in

a

generic

cut-and-project

_sequence c +

_d)

with _{ê, 1/,} d &#x3E;

_0,

it suf-fices to find n E ~ such that d

dn.

Then numbers

_{Ani , An2,}

A(n+1)2

take three different values that are the

_lengths

of tiles in c +

_d).

Proposition

4.3. Let _{ê, 1/} &#x3E; 0 be irrational numbers and let E2 =

[c,

_{c +}

d)

satisf y

- - - - --. - - - -.

The sequence

Ee,l1(O)

has three

types

of

tiles and the

_{lengths of}

these tiles

are

Let us realize that in this case

A*

and

OZ

do not

depend

_{on 17} &#x3E; 0. Therefore also the

_algorithm

_(6),

_{corresponding}

to

_shortening

the accep-tance

_interval,

does not

_depend

on _q

(unlike

the

algorithm

(7)).

The value

of q

influences

only

the

_length

of tiles and not their

_ordering.

Therefore

we can choose the

_{parameter 17}

&#x3E; 0

_according

to our

_needs,

for

_example

we can

_{set 17}

=

c-1

which

corresponds

_to the _case when the

projection

in the

cut-and-project

scheme is

_orthogonal.

Proposition

4.4. Let _{c, 77} &#x3E; 0 and let fl be an mterval

of

length d,

In case that the

_acceptance

window has

length

d &#x3E;

_1,

the

_{parameter 77}

plays

an

_{important role,}

as it is shown

_by

the

_following

_assertion,

which

summarizes the results of this section. Its

_proof

is a _consequence of

algo-rithms

₍₆₎

and

₍₇₎

and uses mathematical induction.

Proposition

4.5. Let _{ê, ’11} &#x3E; 0 be irrational numbers with continued

frac-tion

_[ao,

_ai,

_{a2, ... ~}

and

_~bo,

_{bi , b2 , ...}

_]

_{respectively.}

Denote

_by

and

the sequences

of

the

convergents

associated to - and _’11, that is

o Let 0 d 1. c +

_d)

is a

_{cut- and-project}

_sequences with two

tiles there exist

_{k E No,}

and s E _{1 s ak+l,} such that d = =

1(8 -1)(Pk -

eqk)

₊

Pk-1 -

eqk-ll.

In this case the

lengths

of

tiles are

o Let 1 d. c +

_d)

is a

_{cut-and-project}

_sequence with two tiles there exist

_{k E No,}

and s E 1 s

bk+l’

such that d =

Pk,s =

(S

+

_1)(rk

-I-

_etk)

+ rk-1 -i-

_etk-l.

In this case the

lengths

of

tiles are

Note that we have described all

_acceptance

windows

f2,

for which

has

_only

two

_types

of distances between

_neighbouring

_points.

This

_happens

if the

_length

of n

_belongs

to one of two _sequences of

_{values, Jk,,}

&#x3E;

_1,

6k,s

=

0,

and the

lengths

of

corresponding

tiles increase to oo. For

the other sequence we have

limk-+oo

pk,s =

oo, and the

lengths

of

corresponding

tiles tend to 0.

5.

_Complexity

of

_{cut-and-pro ject}

_sequences

It is useful to introduce a function that allows to determine the

_neighbour

of a

_{point x}

E

_according

to the

_position

of ~* in f2. Its definition is based on

Corollary

4.2.

Definition 5.1. Let Q =

[c,

_{c +}

d)

be _a bounded

interval,

and let

Ai , A2,

and

_O1

+

_A2

be the tiles in

_E,,,7(Q),

such that

_A*

&#x3E; 0 &#x3E;

_il2.

Let

The function is called the

_stepping

function of

_EE~~(S2).

If there is no

misunderstanding

possible,

we shall omit the

subscripts.

The _map

_f

is

_actually

an

_exchange

of three intervals.

_Codings

of

three-interval

_exchange

transformations are studied for instance in

[11].

The

three-interval

_{exchange corresponding}

to the _map

_f

from the above defini-tion is associated with the

_permutation

_{(3, 2,1).}

_Every

_exchange

of three intervals with

_permutation

_{(3, 2,1)}

can be realized as a _map

Note that the

_stepping

function is

obviously

invertible. The

_right

neigh-bour of x E

_Ee,1J(O)

is

_{~fE,~(x*)~ *.}

_Similarly,

the

_n-tuple

of its

_right

neighbours

is

_{given by}

_{( f (x*)) *,}

_{( f ~2&#x3E;(x*)) *,}

... ,

( f ~"~(x*)) *.

In

par-ticular,

for _any

_{given Yo EOn}

_{7G(e) we}

have

Let us denote the tile

_O1

+

_A2

_by

letter

_A,

the tile

_O1

_by

letter B

and the tile

A2

by

letter C.

_According

to the first n

_{right neighbours,}

we can associate to _every

_point

E a word of

length

n in the

alphabet

A =

IA, B, Cl.

This word will be denoted

by

word (x, n).

For

xi, x2

E

_-1

the words

_{word (xl, n)}

and

_{word(x2, n)}

are different if and

only

if there exists an i =

1, 2, ... ,

_n, such that at least _one

discontinuity

point

of

_f

lies between and

_f(i-l)(x2).

Therefore the set of all

points

x* E

_satisfying

_word(x,

_n)

=

w, has the form where

by Qw

and without loss of

_generality

we shall consider 9 semi-closed. In

particular,

we have

It is known that the

_density

of

_points

in a

cut-and-project

set is

proportional

to the volume of the

_acceptance

window

_[19].

It follows that the

_density

of the word w is

_proportional

to the

_length

of interval

_f2,,.

In

_particular,

we

have the

_following

_proposition.

Proposition

5.2.

_{Let e, q}

be irrationad _-"1, and let S2 be

a bounded interval. Let w be a

factor

in the word Then

for

the

density

ow

of

the

factor

w we have

In

_particular,

_for

the densities

_of

letters

_{A, B,}

C we have

Recall that the

_complexity

_C(n)

of an infinite word u is the number of

different factors in u of

_length

n. In our case it is

_{given by}

the number of all

_non-empty

intervals

_Qw

that cover Q. The number of such intervals is

determined from the number of left

_end-points

of these intervals. A

_point

z is a left

_end-point

of a

_non-empty

interval

_Qw

if and

_only

if it is either a left

end-point

of 0

itself, i.e., z

=

c, or there exists

i E f 0,1, 2 ... n - 1 ~

such that

_{f i (z)}

is a

_{discontinuity point}

of the function

_{f ,}

_i.e.,

fi (z)

_E

{c

+ d -

_{0 i ,}

c -

~2}.

Therefore we have the

following prescription

for the

complexity,

where we denote for

_simplicity

of notation a = _{c +} d -

ili

and

Q

= _{c -}

~2.

We can now determine the

_complexity

of c +

_d).

Theorem 5.3. Let C denote the

_{complexity ,function}

+

_d).

. then

9

If

d _E

Z[e],

then there exists a

_unique

_{non-negative k E No}

such that

f ~k~ (a) _

(3

or

(p)

= _cx, where

a, Q

_are the

discontinuity

_points

Proof.

and are

_stepping

functions

corresponding

to

_acceptance

intervals 92 and f2 + t

_{respectively,}

then =

t)

_{+ t} for

any translation t of the interval 11. This means that blocs

occurring

in the word

c+ d)

occur also in

c+d+t)

and vice versa. Two words with

such

_property

are said to

_belong

to the same local

isomorphism

class. The

words and must have the same

complexity.

Without loss of

_generality

we can thus consider f2 =

[0, d).

Recall that

~i,

~2

E Z[e],

and that the

_{discontinuity points}

of

_f

for such an interval

S~ are a = d -

A*

and

{3

=

-A*.

Our aim is to determine the number of

elements in the set

The function

_{f ~~~}

has no fixed

_point

for _any

_k,

which

implies

the

following

facts:

(i)

The

_points

a, f ~-1~ (a), ... , f ~-~n-1» (a)

are

_mutually

distinct. The same holds if we

replace

a

_{by Q.}

(ii)

One can never have

_{simultaneously}

_,8

and

f’(,8)

= a for some

_{k, i}

E N.

(iii)

Since

_f((3)

= 0 = _c, the

point

0 is the closest

right

neighbour

of

(3-*

and we have

f -~ (,~)

~

c for all i E

No.

~

_Suppose

that d 0

Z[c].

Then a = d -

Z[e]

and (3 =

-A*

_E

Z [,-].

Since

_f(y)

E if and

_only

if _y E we have

f ~~‘~~ (a) ~

{O, {3,

f ~-1~

(a)7....

¡( -en-I»~ ((3)}.

Together

with

_properties

_(i)

and

(iii)

this means that the set M has 2n + 1 elements.

~ Let now d E

_{?~ ~~J,}

i.e.,

the star map

images

a-*, (3-*

of both

discon-tinuity points

of

_f

and the

_point

0

_belong

to

_E~~,~(S~).

Since the

_point

0 is the closest

_{right neighbour}

of

_(3-*

we have either a-*

~i-*

or

0 a-*. In the first case it means that there exists a

_non-negative

k such that a-* is the k-th left

_neighbour

of

_(3-*,

i.e.,

# a

for i k and

f ~-~} (,~)

= _a. In

the

second _case that there exists

a

_non-negative

k such that 0 is the k-th left

_neighbour

of

a-*, i.e.,

f t-i~ (a) ~

{O, {3}

for i k and

_/(-~)(~)

= 0 =

f((3).

Starting

from th value n -1= k the

_pairs

of elements in the set

_coincide,

which

influ-ences the

_cardinality

of this set and

_consequently

also the

_complexity

of the infinite word.

0

Since infinite words with

_complexity

n+ const.

_{(quasisturmian}

sequences,

see

[8])

are well

described,

we shall now focus on words of

complexity

2n+1.

According

to the result of Alessandri and Berth6

_{[2, 7J,}

for a

_{given n}

E N the densities of factors of

_{length n}

can take at most

_{3 (C(n}

_{+ 1) -}

_C(n))

= 6 values. In

_fact,

there are

only

five

frequencies

for factors of

given

length.

This can be

_proved

_following

the ideas in

[2]

and

_using

the fact that the

words associated to

_{cut-and-project}

_sequences are

codings

of three interval

exchange.

Let us determine the maximal in- and

outdegree

in the

Rauzy graphs

corresponding

to the infinite words in consideration. Let e be a factor of

length n

+ 1 in the infinite word

_u,,,7[c,

c +

_d)

and let v be the

prefix

of e of

length

n.

Obviously

S2e C 011.

Therefore in the

Rauzy graph

rn

the vertex v is the

_{starting point}

of the

_edge

e. If

52e

=

011,

then the

outdegree

of the vertex v is

equal

to one. This

happens

only

if the

points

f (-n) (a),

f ~-n~(~3)

do not

_belong

to the interval

_011.

If the interval

₀₁₁

contains

_exactly

one of the

_points

f -"

(a),

f -" (,Q),

then

f2v=

52e1

where el, e2 are the

only

two factors of

_length

n + 1 with

_{prefix v}

and the

outdegree

of the vertex v is

_equal

to 2. In case that both

points

f (-n) (a),

f ~-"~

(p)

happen

to

_belong

to the interval

_011,

the

_outdegree

of the vertex v

is

_equal

to 3 and all other vertices in the

_Rauzy

_graph

_rn

have

_outdegree

1. Since the set

{ f ~-"~ (a), f ~-"~ (~i) ~ n

E

_N}

covers the

_acceptance

interval

SZ

_densely

and

_uniformly

_[22],

the

_lengths

of intervals

₀₁₁

tend to 0 as n tends to

_infinity.

_{If for every n} E N both of the

_points

_{f (-n) (a)}

and

f ~-n~ (/~)

fall into the same interval

011

for some factor v of

_length

_n, then

= 0. Take c

large

enough

so that

I f (n) (a)

-f -’ (,)

I

j. From the

_properties

of the

_{stepping function,}

as

_piecewise

linear function with

_slope

_1,

it can

_happen

that

I =

I f ~-n~ (a) - f ~-"~ (,Q) I

.

However,

this cannot be true for every n. In the

opposite

case, we have

f ~-n-1~ (a)

I

»

I f~-n&#x3E; (a) -

f~-n~ (,(3)

I

and thus

_points

_{f ~-’~-1~ (a),}

_{f ~-n-1~ (/3)}

cannot be in the same interval

011.

Thus we have

_proved

the

_following

statement.

Proposition

5.4. Let c, 77 be irrational

numbers,

20137?~ and let S2 =

[c,

c +

_d)

be a non

_empty

interval. Then there exists an _no such that

for

all n _{&#x3E; no} the maximale

_indegree

and the maximale

_outdegree

in the

_Rauzy

graph

rn

of

’UE,1J(O)

are

_equal

to 2.

6.

_{Geometrically}

similar

_{cut-and-project}

_sequences

Two sets A and

A

C R are

geometrically

_similar,

if there exist

p, 1b

E

_R,

cp =1=

0,

such that

A

=

OA

+ ~.

We shall denote this

_property

by

A 19

A.

Remark 6.1. Note that if two

_{cut-and-project}

_sequences and

are

_{geometrically}

similar with a

positive

similarity

factor _cp, then

the

_{corresponding}

infinite bidirectional words and coincide.

A converse statement is not

_true,

cf.

_Proposition

4.4.

In order to describe classes of

_mutually

_{geometrically}

similar

We have the translations

7~2

_{+ ( g )}

=

Z~,

for

a, b

_E

Z,

and the

group of rotation

_symmetries

of

?~2.

_They

are

_{given by}

all

integer

valued matrices A

with determinant + I . Then

AZ

=

72.

These

symmetries

of Z2

correspond

to the

_following

transformations of

_{cut-and-project}

_sequences.

Proposition

6.2. Let _{e, r~} be irrational 2013y?~ and let S~ be a

bounded interval.

~ For x we have

o Let _a,

b,

_c, d be

integers,

such that ad - bc = Then

Proof.

First,

let x = _{a E}

Z[77].

Then

Now

_{let A = ( g [ )}

be an

_integer

valued matrix with determinant ±1. Then

The _group of

_integer

valued matrices with determinant :J:l is

_generated

by

matrices

_{Ai =(~),A2}

_{= ~ o -°i ~ ~}

_{A3 =}

_{(~ å ).}

_Elementary

transforma-tions of

_{cut-and-project}

sequences for these matrices are

Using

the above

_elementary

transformation we show that we _may limit our considerations to c, 77 and 11 with certain

properties,

without

loosing

any infinite bidirectional word. The aim of this section is to prove the

following

theorem.

Theorem 6.3. Let _{e, q} are irrationat

numbers, - :A

_2013~. Let Q be a bounded

interval. Then there exist irrational, numbers

_6,17

and an interval

fl,

such

that

The

_proof

of the theorem will be divided into two lemmas. Before

_stating

the

_lemmas,

let us recall certain

_properties

of continued fractions

[12].

If ~

&#x3E; 0 is an irrational number with continued fraction

[ao, ~i,~2! -"

]

the sequence of the

convergents

associated

_{to ~,}

then for _every

Lemma 6.4. For any irrational El

~

-77, and bounded interval

í2,

there exist irrational numbers

_{~, ~}

and a bounded interval

S2

such that

Proo f .

Using

transformation

₍₁₂₎

we _may assume without loss of

_generality

that 1] ’ &#x3E; 0. We first find

_convergents

_q2n

and associated to _77, so that

q2n 1 "

2013c [S2s.

From the

_property

_(i)

of continued fractions we have

’

Let us define the matrix A =

( a b )

= for

transforma-c d -92n-E-1 9’2n

tion

_{( 10) .}

Then from

₍₁₄₎

we have

and from

₍₁₅₎

we have

From the

_property

_(iii)

of continued fractions we obtain

In the transformation

₍₁₀₎

with the matrix A we obtain a new

_acceptance

window Ô

:= Q. Let us find an estimate of the denominator in the

fraction

Note that in the estimate we have used

_property

(ii)

of continued fractions.

Since +oo and

₁₁₇

+

_-1

_~

_0,

it is

_possible

to choose a

sufficiently

large

n, so that the

_length

of the interval

f2

is smaller or

_equal

to 1. Cl

Lemma 6.5. Let _{ê, 17} &#x3E;

_0,

_ê,17 irrational and let S2 be an interval

of

length

1. Then there exist irrational numbers

_{~, ~}

and an interval

f2

satis-lying

Proof.

Without loss of

_generality

we _may.assume that - 1. Otherwise we

would use the transformation

(11)

to

_get

where - -

_[e]

1. We first show that there exist

_{0 ~ 1, ~ &#x3E; 0}

and

C2

such that

- -

-The case of

_{JOI I}

&#x3E; e is

_trivial,

because it suffices to

_{put t = e, ij =}

_1/

and

C2

= Q. Assume that for

given

f2 _we have

Inl

e. Let

(~-) ~

be the

_convergents

associated to e.

According

to

_property

_(ii)

of continued

fractions,

the _sequence

_{(IPn -}

is

_decreasing

and thus it is

_possible

to find n E N so that

Let us transform the

_{cut-and-project}

set

_using

the

transforma-tion

₍₁₀₎

with matrix A =

( qn_ 1 qn ) ~

We obtain

E~,~ (SZ),

where

According

to

_properties

_(i)

and

_(ii)

of the continued fractions we have

0 t 1. The

_{number 7y}

is

_clearly

_positive.

Since for

_Inl

we have

(16),

for

the

_{length of C2}

we _may write

- I IV ,..

what was to be shown.

If moreover

101

I

&#x3E; 1-

_ë,

we _may set % _

ë, f¡

:=

_~,

and f2 :

=

C2,

and the

lemma is

_proved.

It thus remains to solve the case

This, however,

is

_{possible only}

_1,

_i.e.,

its continued fraction has the form t =

_[0,

cl , c2, ...

with

2. Since

it is

_possible

to find a minimal s

6 {1,2,...,

cl -

1}

so that

Now we use the transformation

(10)

with the to

_get

Let us

verify

that

_6,7y

and

S2

satisfy

the

inequalities required by

the lemma.

The

_{paralneter fi}

is

_positive

as it is a ratio of

_positive

numbers. For e we use the

_inequality

(18)

to obtain 0 e 1. For the estimate

of Ö

we derive

from

₍₁₇₎

and

₍₁₉₎

that

In order to

_complete

the

_proof

of the

_lemma,

it remains to show tha

if2i

&#x3E; 1 - e. If the minimal s satisfies 5 _{cl -}

_2,

we have 1 -

(s

+

_l)e

1 - st which

_implies

- --

-If the minimal s is s =

ci - I,

then

automatically e

_&#x3E; 1 - ê

which,

together

with

₍₂₀₎

_implies

in both cases 1 &#x3E;

if2i

&#x3E;

_{max(ê,1 -}

_e)

and _proves the

lemma. 0

Remark 6.6. Note that in Theorem 6.3 we _may choose the

interval f2

in

such a _way that the

similarity

factor between and is

pos-itive,

thus infinite bidirectional words associated to these

_{cut-and-project}

sets are identical. If it

happens

that = for _{some p}

_0,

we can use the transformation matrix A =

(Õl ~1)

_to obtain

A I1

Therefore

_using

the interval

-s1

instead

of Ö

_gives

us a

positive

similarity

factor _-Sp.

7. Combinatorial classification of

_{cut-and-pro ject}

sequences Let us recall that two words u =

(un)nez

and u =

(fill),,EZ

in

alphabets

,r4 and

4,

_{respectively,}

are

combinatorially equivalent,

if there exists a

bi-jection

h:

A - /l

and an _{no E} Z such that

_{fin =}

for all n E Z. We denote this û. Note that if a

bijection h

should

exist, certainly

the

_density

of a letter a in the word u must be the same as the

_density

of

h (a)

in Û.

Remark 7.1. Recall that two words associated to

_{cut-and-project}

sets

which are

_{geometrically}

similar with

_{positive similarity}

factor are

combi-natorially equivalent,

see Remark 6.1.

_Using

Theorem 6.3 and Remark 6.6 we _may thus limit our

consideration,

without loss of

generality,

to words

with

_parameters

_satisfying

and thus we have

For the densities of tiles

_{A, B,}

and C we have in this case

respectively,

cf.

_Proposition

5.2.

Proposition

4.4 shows that two

_{geometrically}

non-similar

_{cut-and-project}

sequences may

correspond

to

combinatorially equivalent infinite

words. Let

us show

_yet

another

_example

of

_{combinatorially}

equivalent

words.

Example

2. Consider Sturmian words _u, u in the

_alphabet

_{10, 11}

with the

slope

a, and a = 1 - a

_{respectively,}

where 0 a 1.

Obviously, u

and t are

combinatorially equivalent

since fi arises from u

by

replacing

O’s with

l’s and vice versa. Such Sturmian words

correspond

to

_{cut-and-project}

sequences with Q a semi-closed interval of

length

1,

see

Example

1. For

77 &#x3E; 0 we have

Both of the sets have two tiles of

_{lengths 77 and 77}

+ 1. More

_precisely,

the distances are

We can see that

The above

_example

illustrates that for

_acceptance

window of

_length

1 the

interchange -

++

_{1 -,-, fl}

++ -0

_gives

_{combinatorially equivalent}

words. The

following

lemma states that the same

_property

holds for

general

_acceptance

window as well.

Lemma 7.2. Let

_{0 e 1, ~ &#x3E; 0 be}

irrational

_numbers,

and let S2 be an

interval

_{of length}

_{max(e,1 - e)}

1. Then

, , , . , , ..

Proof.

Although

=

7G (1-

eJ,

the star _maps from

7G r)

to

Z[e]

=

are different for the

_{cut-and-project}

_sequences

_{Ee,l1 ( -0)}

and

_E1-e,17(0).

We shall therefore

_distinguish

them

_{by indices,}

According

to

_Proposition

4.3 in both

_{cut-and-project}

sets the three dis-tances are _q, 1

_{+ 1/}

and

1 + 2r¡.

Their

images

under the star map are however

different for * 1 and *2,

namely,

For the

_proof

we use two facts:

Let Then the fact

_(F2)

_says that the _sequence of distances to

the left from the

_point

in the set

_{E~~~ (SZ)}

is the same as the sequence of distances to the

_right

from the

_{point - xo * 1}

in the set

_{E ( - SZ ) .}

The fact

_(Fl)

_says that the _sequence of

_steps

to the

_right

from the

_point

in the set is the same as the _sequence of

_steps

to the left from the

_point

_{xo *1 in}

the set

_E~~~(SZ),

_only

the

_length

of

_{steps 77}

and 1

_{+ ~}

are

_{interchanged.}

0

Note that the

_density

of the shortest tile 77 in the set

_{EE~,~ (-S~)}

is

_equal

to

1- a,

whereas the

_density

of the shortest

_{tile 77}

in the set is

_equal

to 1 - It is therefore obvious that the

_{cut-and-project}

sets

and are not

_{geometrically}

similar.

_{Nevertheless,}

the associated infinite bidirectional words are

_{combinatorially equivalent.}

From _every Sturmian sequence in the

alphabet

~A, B}

we can form a

special

word in the

_alphabet

_B,

_C}

in the _way that in between _every

two letters of the Sturmian sequence we insert k-times the letter C. Such a _sequence we call a

_k-padded

Sturmian _sequence.

Definition ?’.3. Let u be a Sturmian word in the

_alphabet

(A, B).

Let

k E N and h be a

B } -~ IA, B, C } * ,

_{given by}

h (A)

=

ACk ,

h(B)

=

BCk.

The word

h(u)

is called a

_k-padded

Sturmian word.

The

_following

_proposition

shows that

_k-padded

Sturmian sequences can

also be

_{cut-and-project}

_sequences.

Proposition

7.4.

_{Every k-padded}

Sturmian word is a word associated to

a

cut-and-project

_sequence.

Proof.

Let u be a Sturmian word with

_slope

_cx, 0 a 1.

Similarly

as in

Examples

1 and 2 we can consider u to be the word c +

_1),

for some c E R

_{and p}

&#x3E; 0. For a

_given

k e N we form a

k-padded

Sturmian _sequence

and want to find

parameters

~, TJ,

SZ,

so that the

k-padded

Sturmian word

is a word We

_put

Note that f2 can be written as 0 =

[c,

c ₊

j)

for c = _cc

and i

=

(k

₊

1)e.

Since

₁

c

1

we have

i

1 and therefore star

k+2

images

of tiles in are

_,1-

_-, and 1 - 2e.

For such

_parameters

the

_stepping

function is

_{given by}

We can

_verify

_easily

that

Therefore every letter A or B is followed

by

the

_string

of J~ letters

C,

whereas the

_(1~

+

_1)-th

letter is different from C.

_Moreover,

the restriction of

to OA U

Q B

is a scaled

_stepping

function

_{f a ~ ~ 1.}

0

Theorem 7.5.

. For _any irrational numbers _ê,

~

_-77, and a bounded interval 0

there exist irrational t and an interval

Ö satisfying

such that

_u,,,7(ft)

_for

any

irrational

&#x3E; 0.

.

0 Let _e,

_{t, q, fi}

be

_positive

irrational numbers such that 0

_{e, e}

2

Let

_f2,

f2

be intervals

_satisfying

1 - E

_1,

1-

IÖI

_1,

u S c u

,.

Then

-Then

,

-

either 6 = e and there exists _{x E}

Z[e]

such =

Z[e],

i. e.,

=

-

or there exists

_{k E N,}

and an irrational _a, 0 a

_1,

such that

is

_k-padded

Sturmian word with

slope

a and is a

k-padded

Sturmian word with

_slope

1- a.

The

_proof

of the first statement of the theorem is a direct _consequence

of Lemma 7.2 and Remark 7.l. In the

_remaining

_part

of the section we are

going

to _prove the second statement.

Let us determine

_using

the

_stepping

function the

_possible

_lengths

of blocks of one letter in the word Let us denote

by

min(A), max(A),

(resp. min(B), max(B))

the minimal and the maximal number of letters