An alternative construction of normal numbers

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

E

DGARDO

U

GALDE

An alternative construction of normal numbers

Journal de Théorie des Nombres de Bordeaux, tome

12, n

o

_{1 (2000),}

p. 165-177

<http://www.numdam.org/item?id=JTNB_2000__12_1_165_0>

© Université Bordeaux 1, 2000, tous droits réservés.

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

An

alternative construction

of normal numbers

par EDGARDO UGALDE

RÉSUMÉ. Nous construisons une nouvelle classe de nombres nor-maux en base b de manière récursive en utilisant des chemins

eulériens dans une suite de

_digraphes

de de

_Bruijn.

Dans cette construction

_chaque

chemin est

_fabriqué

comme une extension du chemin

_précédent,

de telle manière _que le bloc

_b-adique

déterminé par le chemin contienne le nombre maximal de sous-blocs

b-adiques

distincts de

_longueurs

consécutives dans

_{l’arrangement}

le

_plus

compact. Toute source de redondance est évitée à

_{chaque étape.}

Notre construction récursive est une alternative à

_plusieurs

cons-tructions par concaténation à la

Champernowne

qui sont bien

connues.

ABSTRACT. A new class of b-adic normal numbers is built

recur-sively by using

Eulerian

_paths

in a _sequence of de

_Bruijn

_digraphs.

In this _recursion, a

_path

is constructed as an extension of the pre-vious one, in such way that the b-adic block determined

by

the

path

contains the maximal number of different b-adic subblocks of consecutive

_lengths

in the most _{compact arrangement.}

_Any

source of

_redundancy

is avoided at _{every step.} Our recursive

con-struction is an alternative to the several well-known concatenative constructions à la

_{Champernowne.}

1. INTRODUCTION

Let b be a fixed

_integer.

A number x E

_{[0, 1]}

is a b-adic normal if each

block

_{q(1)q(2) - - -}

_q(k)

on b

symbols

_appears in the b-adic

_expansion

of x

with

_frequency

b-1. In

_[Bo]

Borel

_proved

that the set of all b-adic normals is

a set

_{of Lebesgue}

measure

_1,

but it was

_only

35 _years after Borel’s

_proof

that

Champernowne

[Ch]

presented

the first

_explicit

normal number in base 10. He

_proceeded

to concatenate the decimal

_expressions

of all natural numbers

to

_yield

the decimal

_expansion

0.123456789101112131415161718 .... Several other constructions of normal numbers have been

_proposed

fol-lowing

Champernowne’s

idea, i.e.,

a concatenation of blocks of

digits

of

in-creasing

length.

Besicovitch

_[Be]

_proved

that

_{concatenating}

the _sequence of

squares of all the natural numbers

_produces

the normal number 0.149162536 496481100121 ~ ~ .

_Copeland

and Erd6s

_[CE]

_proved

that the concatenation of all

_prime

numbers

_gives

the normal 0.23571113171923293137 .... More

recently

Schiffer

_[Sc],

and Nakai and Shiokawa

_[NS]

_proved

that for a

non-constant,

eventually

increasing

polynomialp,

the number

[p(4)] -"

is also a normal number. Here

~x~

stands for the b-adic

_expression

of the

_integral

_part

of ~. This construction

produces

both Besicovitch and

Champernowne

numbers.

In this paper we

_present

a recursive construction of normal numbers

which is not a la

_{Champernowne.}

Our b-adic

_expansions

are

produced

through

Eulerian

_paths

in a _sequence of de

_{Bruijn digraphs}

of

_increasing

size.

_Thus,

we call the numbers

_{generated by}

our

_algorithm

Eulerian normal

numbers.

The _{paper is}

_organized

as follows. In Section 2 we

_give

the main

defi-nitions. The

_algorithm

to

_generate

Eulerian normal numbers is

_presented

in Section 3. This section is

_mainly

combinatorial. Section 4 is more of a

probabilistic

nature. There we _{prove that} Eulerian numbers are indeed

normal and we

_give

an estimate of their

discrepancy. Finally,

the rate of

convergence to

equidistribution

for the Eulerian numbers is discussed in

Section 5.

2. DEFINITIONS AND NOTATION

2.1. Normal numbers.

_Except

for the rationals of the kind

_m/b’,

a

num-ber x E

_0,1

has a

_unique

b-adic

_expansion

x(1)(2) - x(n) ,

which is

the _sequence on

Zb

such that x = The b-adic block from site

n to site n + k in the b-adic

_expansion

of x is denoted

_by

_{x(n : n + k).}

A number x E

_{[0, 1]}

is said to be a b-adic norrraal number if for each

finite b-adic block

_{q(l : k)}

E

Zk

where

_{N), q(l : k))}

is the

_frequency

of

_{q(l : k)}

as subblock of

~(1 : N).

The rate of _convergence to

_{equidistribution}

is measured

_by

the

_(N,

k)-discrepancy

associated to x E

_{[0, 1].}

It is denoted

_by

_Drr,k

and is defined as

follows:

We

_readily

see that x E

_~0,1~

is normal if and

_only

if 0

for all k E N.

2.2.

_Digraphs.

A

_digraphs

is a

couple

(Tl, A),

where V is a finite set and A a

_binary

relation on that set. The elements of V are called

_vertices,

and

the ordered

_pairs

in A are called arrows. We denote

_{by v H}

v’ the arrow

E A.

To a

_given

vertex v E V we associate the sets

_{0(v) =}

_fv’

E

_v’l

and

_{I(v) =}

_{v’

E

_vl.

Given v H v’ we call v’ the head and v the tail.

A

_{digraph (V, A)}

is

_b-regular

if for each vertex v E V

_{I I(v) 1 0 (v)}

_I

= b.

A

_path

in

_{(V, A)}

is a _sequence

(vl,

_V2,

... , v9)

of vertices such that for

each 1 i _{s, vi H vi+l.}

A

_path

_{(Vl,V2,... va)}

in

_{(A, V)}

such that Va M vi defines a

cycle

in

(V, A),

which is the

_equivalence

class of all

_paths

of the kind

_(Vi,

_{vi+i, ... , vs,}

vl, ... ,

vi- i)

obtained

by cyclic

permutations

of the vertices in the

_original

path.

Since a

_cycle

is determined

_by

one of its

_{possible paths,}

we use _any of its

paths

to denote it.

The

_digraph

_{(V, A)}

is connected. if for each

_couple

_{v, v’}

of vertices there

is a

_cycle

(v,

_v,....

v’,

vi, ... ,

_containing

them.

Let

_{(V, A)}

be a connected

_digraph.

The arrow v H v’ in A is a

bridge

if the

_digraph

_{(V, A B}

_v’~),

obtained

_{by erasing}

that arrow form the

original digraph,

is no

_longer

connected.

A

_path

where each vertex in V appears

exactly

once is a Hamiltonian

path,

while a

_path

where each arrow _appears

_exactly

once is an Eulerian

path.

Correspondingly,

Hamiltonian

_cycles

and Eulerian

_cycles

are

_cycles

made

_respectively

of Hamiltonian and Eulerian

_paths.

2.3. The de

_{Bruijn digraphs.}

The

_(b,kJ-de

_Bruijn

_digraph,

denoted

_by

B(b, k),

has vertices in the set of b-adic blocks of

_length

_k,

and arrows

between

_overlapping

blocks.

_Thus,

_{p(l : k)}

~-4

_{q(l : k)}

if and

_only

if

p(2 :

k)

=

q(l :

k -

1).

Note that

B(b, k)

_is

_b-regular

for _{all k.}

The

_family

_{B(b, k)}

was introduced

_by

de

_Bruijn

[Br]

and was

_already

used in the framework of normal numbers

_by

Good

_~Go~.

An Eulerian

_path

on

B(b, k)

defines a Hamiltonian

path

in

B(b, k

+ 1)

by

identifying

the arrows of the first

_digraph

with the vertices of the second

digraph.

We also associate

_paths

in

_B(b,

_k)

to b-adic blocks. The

_path

(a(1 : k), a(2 : k

+

_{1), ... , a(s : s}

+ k -

₁₎₎

in

_{B(b, k)}

is associated to the block

_{a(1 :}

s + k -

₁₎

E

~b+k-1,

A b-adic block

_{q(1 :}

_s)

of

_{length s}

&#x3E; t is an extensions of the b-adic block

p(1 : t) if p(1 : t)

=

q(l : t).

Similarly,

an extension of a

_path

(vl, v2, ... , vt)

is a

path

of the form

(vl , v2, ... ,

_{vt, ... ,}

vs ) .

3. CONSTRUCTION OF THE EULERIAN NUMBERS

A b-adic Eulerian number contains in its b-adic

_expansion

all the b-adic blocks of all

_lengths,

but in the most

_compact

_possible

_arrangement.

The main

_ingredient

in the construction of Eulerian numbers is the func-tion Ext which extends b-adic blocks associated to Hamiltonian

paths

to

blocks associated to Eulerian

_paths.

For b &#x3E;

_2,

_given

the b-adic block

a(l : bk + k -

1)

defining

a Hamiltonian

_path

in

B(b, k),

k

-1))

=

a(l :

₊

k)

is _one of the

_possible

extensions of

a(1 : bk

₊ k -

1)

whose associated

_path

in

_B(b,

_k)

is Eulerian.

Since there is more than one

possible

extension with the

required

prop-erty,

Ext is a non-deterministic function.

In the 2-adic case this extension is not

_possible,

but a similar

algorithm

can be

_{implemented. Nevertheless,}

since 2-adic normal numbers can be

obtained from 4-adic normals

_(see

_[Ma]),

in what follows we

only

examine

the case b &#x3E; 2.

To construct and determine a b-adic Eulerian number e =

we

only

have to iterate the function

_Ext,

A

_possible

outcome of this

_procedure

_gives

a number whose 3-adic

ex-pansion

starts with

We call _any number whose lr-adic

_expansion

is obtained in the _way de-scribed

_above,

an Eulerian number.

3.1. The non-deterministic function Ext. To

_complete

the

_description

of this

_{construction,}

we have _{to prove} that these extensions exist. In fact we will _propose an

_algorithm

to construct the

_outputs

of the function

_Ext,

following

the idea in

_[Tu],

where the

_following

result is

_proved.

Theorem 1. A

_digraph

_{(V, A)}

has an Eulerian

cycle

(vo, vl, ... , I VI, vo) if

and

_{only if}

_{(V, A)}

is connected and

_for

each vertex v E V

_{IO(v)1 = II(v)l.}

Note that the de

_Bruijn

_{digraphs satisfy}

the

_hypotheses

of this

_theorem,

but in order to extend a Hamiltonian

cycle

into an Eulerian one we need a

little more than this result. We _prove the

following.

Theorem 2. Let

_{(V, A)}

be a connected

digraph

with no

bridges,

and such

that

_for

all v E V

_{10(v)l =}

_N(v).

_Suppose

_(A,Y)

admits a

Hamiltonian

_{cycle defined by}

the

_path

_{(vo, vl,}

... ,

7 vi).

Then there exists an

Eulerian

_cycle

determined

_by

a

path

(VO,V1,...

ve)

which extends

the

_given

Hamiltonian

_path.

We divide the

_proof

into two

_parts.

First we

_{just apply}

the

_algorithm

skeleton of the construction. After that we

_modify

the Eulerian

_cycle

we

obtained to

_give

it the desired form.

Proof.

Let us order all the arrows of A in a table.

In this table there is one line for each vertex in V. In the line

_{corresponding}

to the vertex v we

_put

all the arrows of the set

_O(v)

_following

an

arbitrary

order,

except

for the last

_entry,

where we

place

the arrow

_appearing

in the

Hamiltonian

_cycle.

Since

_{(V, A)}

is not

_{necessarily regular,}

different rows

may have different

lengths.

The initial Eulerian

_{cycle: Using}

this table we determine an Eulerian

cycle

as follows.

~ Initialization: _{vl is the first vertex} of the Eulerian

_path.

~ Recursion: At a

_given

_step

of the construction we

already

have a

sequence

(Vi,... , v).

Let v - v’ be the first arrow

_appearing

in the

line of the vertex v in the table.

-

Increase _{the sequence}

_{(vl,... v)}

_{by appending}

v’.

_So,

the new

sequence is

(v,~, ... ,

v,

v’ ) .

-

Erase v H v’ from the table.

This

_procedure

_stops

at a vertex v when all the arrows of

O(v)

have been

erased from the table. When it

_happens,

the

_resulting

_path

... ,

v)

has

N(v)

arrows with v as

_tail,

and in the last arrow of the

_{path v}

is head.

Since

_10(v)l

=

BI(v)l,

the

only

possibility

is _v =

vl.

Now,

let us show that all the arrows of A are included in this

_path.

Suppose vk

H v’ is not in

_{(Vi,... , 7 VI);}

then neither is Vk ~ _Vk+1, which is

the last arrow in the line of _vk in the

_original

table. Because of the choice

we made for the last entries of each line of the

_table,

this would

_imply

that

none of the arrows _{Vk+1 M Vk+2, vk+2 H Vk+3,... , Vi-1} M _vl is in the

_path,

but this is

_impossible

since the line

of vi

was

already

erased from the table.

The sequence

_{(vi, ... , vl)}

determined

by

this

_procedure

defines an

Euler-ian

_cycle

which _preserves the order of the arrows in the

_original

Hamiltonian

In this

_picture,

either

_Ck

is

_empty,

or it is a

_path

(Vk,l,

_{Vk,2, ... ,}

which

_{preceded by vk}

defines a

cycle

7 Vk,nk).

We denote

this

_{cycle by vk fH}

_{C~ .}

Modifying

the Eulerian

_cycle:

The

_{following algorithm}

allows us to move

the

_cycles

vk -

Ck

for k

_t,

to include them inside the

_{cycle vi}

H

_Ci.

Note that for 1k

_t7

if

_Ck

is not

_empty,

then it

_only

contains vertices with index

_larger

than or

equal

to k. This is because when we first enter

Ck,

we

have

_already

finished all the lines associated to _{vertices vj with}

_{0 j}

k. If

_Ck

shares a vertex with

_Oi

then we can move

Ck

to include all its

arrows inside a new

Cl.

. Basic IteTation: For k = 0 to l -

1,

if

Ck

is not

empty,

and

M

_Ck

shares one vertex

_{with vi -}

_Cl,

then

_modify

the Eulerian

_cycle

as

indicated below.

Let vk -

Vk,l t-+

... _M

Vk,r r-7

~ ~ ~ H _{vk,nk and} Vi ~H

_Ci

_vi H _viI H . - ·

H- Vi s t-+

... · _H vi nl be such that Then erase

Ck

from the

original

Eulerian

cycle

and

replace

Vi ~-4

Cl by

the

_path

where

_Ck

is

_empty

and the new

_Cl

has more vertices than the

original

one.

o Recursion:

Repeat

the

previous

_step

until none of the

remaining cycles

vk -

Ck

shares a vertex

_{with VI.}

H

_Cl.

At the end of the recursion we obtain a

_cycle

which is defined

_by

the _sequence

_{(vo, vl, ... , ,Vi,Ci,Vi).}

This is the desired

extension.

In order to

_complete

the

_proof

we have to ensure that at the end of the

recursion,

which takes no more than i + 1

_steps,

we have eliminated all the

cycles

vk ~

_Ck

for 0 k t.

_Suppose

on the

_contrary

that it remains a non

_empty

_path

_Ck,

with k minimal. In that case all the vertices between

the first vk and the the first vl, have index

larger

or

equal

to k. On the other

_hand,

between the last ve and the first vk, all vertices have index

smaller than k. In this situation all

_paths

_{from vi}

_{with j}

k to vk contain

the arrow _{Vk-1 H- Vk,}

_making

this arrow a

bridge.

This is a contradiction

to one of the

_hypotheses

of the theorem. 0

Extension

_procedure

for the de

_Bruijn

_digraphs.

Theorem 3. Let b &#x3E; 3 and b-adic block

_defining

a

Hamiltonian

_cycle

in

_{B(b, k).}

There exists an extension

e(1 :

bk+1

+

_k)

_of

e(l : bk +

k -

_1),

which

_defines

an Eulerian,

cycle

in

B(b, k).

Proof.

We

_{simply apply}

Theorem 2. Since

_{B(b, k)}

is

_b-regular,

it

imme-diately

satisfies the

_requirement

_{IO(v)1 = II(v)l.}

On the other

_hand,

it is easy to check that

B(b, k)

is connected. The

only

nontrivial

requirement

is

the lack of

_bridges.

Let us _prove that for b &#x3E;

3,

B(b, k)

does not contain any

bridge.

Let us eliminate the arrow _{ala2 ~ - ~ ak -} aZa3 ~ ~ ~ ak+i from

B(b, k).

To

take any

pair

of vertices _{P1P2... Pk} and _{q1q2... qk in}

_{B(b, k).}

For _any

a E

_{f al, ak+1I,}

the b-adic block in

_Zfk,

with a in the middle

repeated k

_times,

determines a

path

in

B(b, k)

which never _passes

_through

the arrow _{ak H a2ag ~ ~ ~ ak+1.} Note that in

the

_path

defined

_by

_{Pkaa... qk,} the arrows

correspond

to

subblocks of

_length

k +

_1,

but the choice of a ensures that all those

sub-blocks are different from _{ala2 ~ ~ ~} Note also that the choice of a _may

be

_impossible

in the case b = 2. 0

The

_procedure

described in the

proof

of Theorem 2 is not

_adequate

for

computer

implementation.

Instead we use the Tutte’s

algorithm:

~ Start with the block

_{e(l :}

_b)

= O1 ~ ~ ~

(b -

1).

~ At _a

_given

_step

of the _{construction we have a b-adic block}

e(1 :

bk

₊

k -

₁₎

which defines a Hamiltonian

_path

in

_{B(b, k).}

-

Erase from

_{B(b, k)}

all the arrows

e( j : j -~ k -1) ~ e( j

+ 1 : j

+

_k)

appearing

in that

_path.

-

Order the arrows of the new

_digraph

in a table similar to the one

we used in Theorem

2,

but _{now, at} the end of each row

_place

the arrows

_appearing

on a

_spanning

tree

_converging

to the vertex + k -

₁₎

_{(see [Tu]}

for

_details).

-

Following

the order in this

_table,

as we did in the

proof

of

The-orem

_2,

we

finally

determine an Eulerian

_{path starting}

at

bk

+ k -

₁₎

and

_ending

at

_{e(1 :}

_k).

-

The b-adic block associated to this

_path,

_appended

to

_{e(1 :}

bk

+

k -

_1),

is the desired extension.

4. DISCREPANCY AND _{EQUIDISTRIBUTION}

Our aim in this section is to prove the

normality

of the Eulerian

num-bers. The structure of the

_proof

is the classical one

_given

in

[Ch, Be].

We use the

_entropic

bound defined

_below,

to estimate the

_{(N, k)-discrepancy.}

Unfortunately

this estimate is not the best

_possible

one. We discuss this

topic

in more detail in the next section.

For all

_{q(l : k)}

E

Z~

and

_{a(l : N)}

E with k N

E N‘,

let

f (a(1 :

N), q(1 :

k))

be the

_frequency

of

_{q(l :}

_k)

as subblock of

a(l : N),

and

Let

be the

_{(e, k)-entropic}

rate of _convergence.

Theorem 4

_{(Entropic bound).}

There are constants a and C

depending

only

on b and

k,

such that

Proof.

Let v be the

_probability

distribution on

7G6

generated by

the b-adic

block

_{a(l : N)}

as follows:

The number of b-adic blocks

_generating

the same distribution can be

computed

as follows.

Let

_k-1)

the

_{"multi-digraph"}

_(where

an arrow _{may appear} several

times

_repeated)

obtained from

_{B(b, k - 1),}

_{by repeating}

each arrow

q(I :

k -

₁₎

~

_{q(2 : k), (N - k}

+

_{1)v(q(l : k))}

times.

By

construction,

this

_{multi-digraph}

allows Eulerian

_paths,

in

particu-lar the

_path

defined

_by

the b-adic block of

_length

N which

_originally

de-termined v. As in the

_proof

of Theorem

_2,

to each Eulerian

_path

there

corresponds

a table

_containing

all the arrows in a

given

order. There are

flq(l:k-1)

(Lq(k)(N - s

+

_{1)v(q(l :}

k))) !

_possible

tables of that

_kind,

not

all of them

_yielding

an Eulerian

_path.

Each Eulerian

_path

in

_{Bv(b, k - 1)}

_corresponds

to a b-adic block

gener-ating

the measure _v, but the converse is not true. All the

_repetitions

of one arrow in

B(b, k - 1)

that we included in

1)

_give

rise to different

Eulerian

_paths

when

_they

are

permuted

in the

table,

but all those

paths

define the same b-adic block. Because of

_this,

the number of b-adic blocks

of

_length

N

_generating

the same distribution v is less than

I I I I

The

_Stirling

_{approximation gives}

us the more useful

_exponential

bound

and

_Ci

a constant

_depending

_only

on b and k.

Zk

On the other

_hand,

the number of

_probability

distributions in

_Zj

that

can be

_{generated by}

a b-adic block of

_length

N is bounded

_{by C2N",}

where

C2

and a are constants

_depending

_only

on b and k.

From these two bounds we

finally

deduce that

where

_hk(e)

=

I

maxq(l:x)

11I(q(l : k)) -

e)

and C =

CiC2

depends only

on b and k.

It

_only

remains to

_compute

_{hk (e).}

We can do this

by

standard variational

procedures,

taking

into account some

_properties

satisfied

_{by hk}

that are

described in

_[EI].

Those

_properties

allow us to restrict our search of the

minimum to the set of

_product

_probability

distributions in

_Z~.

We obtain

Remark: A similar result can be derived from

general

techniques

of

Large

Deviations

_Theory

_{[El] .}

Here we use a combinatorial

_proof

which follows the

ideas

_presented

in

_[Cs],

where

_they

derive

_entropic

rates from the

_counting

of

_possible

Eulerian

_paths

_given

_by

Tutte’s theorem.

Lemma 1. For all

Proof.

The

_entropic

rate

_{[0, 1 -}

_b-k]

-4 1R is

_smooth,

_convex, and

increasing. Taylor’s

Theorem

_applies,

and

_gives

us the

_quadratic

behavior

From this we

_readily

obtain the result. In

_fact,

since both

_{hk (e)}

and

_{Pk (6)}

=-(bkE/k) 2/(2(b -

1))

are _convex,

they

intersect at most in two

_points.

It is

easy to

_verify

that

_they

intersect

_only

at E =

0,

and for all _{e E}

(0, 1 -

b-k~,

4.1.

_Normality

of the Eulerian numbers.

Theorem 5. E

_(0,1~

be a b-adic Eulerian

numbers,

then

for

all k

The

implicit

constant in 0

_{depends only}

on b and k.

Proof.

Let n E N be the maximal

_integer

such that bn N - n + 1.

_By

construction,

x ( 1 :

bn + n -

1)

contains all the b-adic blocks of

length

n as

subblocks,

and there are of those subblocks

having

a

given

q(l : k)

E

Zj

as

prefix. Then,

On the other

_hand,

the

_frequency

of a

_given

q(l : k)

in + k :

N)

can be estimated from the

_frequency

of

_{q(1 : k)}

inside the blocks of

_lengths

n + 1

_appearing

in

_x(bn

_{+ k :}

_N).

Theorem 4 ensures that if we take _en such that a

then the number of

_{(En, k)-bad}

blocks of

_length

is at most

Under this condition we have

which

_gives

The last term in the

_right

hand side is never

_larger

than

b/(n

+

_1).

Note also that Lemma 1

_implies

that

some

C2

_depending

_only

on b and 1~.

Then,

for n

large

we can choose a

constant

_C,

_depending

_only

on k and

_b,

such that

for all

_{q(1 : k)}

0

As an immediate

_corollary

of this result we obtain the

_normality

of the

5. CONCLUDING REMARKS

The estimate on the

(N, k)-discrepancy

_{given by}

the

_previous

theorem is

not the best

_possible

one. It was shown in

[Sc]

that for a

huge

class of normal

numbers

_including

the

_Champernowne

number 0.123456789101112131415

16 ~ ~ ~ ,

the best bound for the _convergence rate to the

_{equidistribution}

is

O(I / log(N)).

Our numerical

_experiments

show that the

_discrepancy

for the Eulerian numbers should be even

o(I/log(N)),

but at the moment we are unable _{to prove} it.

We also did numerical

_experiments

_using

the

_{generalized Champernowne}

numbers

_(see

_[DK]

for

_details),

and we also found a _convergence

slightly

faster than

Both the

_{generalized Champernowne}

numbers and the Eulerian numbers have in common a random

_disposition

of all the blocks of a

_given

length.

We think this is the source of a faster _convergence.

So,

it is not the efficient

packing

of blocks in the Eulerian

_numbers,

but the random nature of the extension

_function,

that

_produces

a _convergence faster than

We _may think in a

_{probabilistic}

result of the kind: "for almost all Eulerian

number x,

DNk(x)

=

o( 1/ log (N) )".

In order _{to prove} such a statement

we should

_probably

have to

_compute

the distribution of

_DN,k,

considered

as a random variable in and

_apply

a kind of central limit theorem.

Acknowledgements. During

the

_preparation

of this _{paper I} was fellow

of the

_Programa

de

_{Repatriación}

de

_CONACyT

México at IICO-UASLP. I

also received financial

_support

from the Universidad Aut6noma de San Luis Potosi

_through

the Convenio FAI-UASLP No. C97-FAI-03-2.15. I thank Gelasio Salazar and Jesus Urias for their careful

_reading

and comments.

REFERENCES

[Bo] E. _Borel, Sur les probabilités dénombrables et leurs _{applications arithmétiques.} Circ. Mat. d. Palermo 29 _(1909), 247-271.

[Br] N.G. de _{Bruijn, A} combinatorial problem. Konink. Nederl. Akad. Wetersh. Afd. Naturuk. Eerste Reelss, A49 _(1946), 758-764.

[Be] A.S. Besicovitch, The _asymtotic distribution of the numerals in the decimal _{representation}

of the squares of the natural numbers. Math. Z. 39 _(1935), 146-156.

[Ch] D.G. _{Champernowne,} The Construction of The Decimals Normal in base Ten. J. Lond. Math. Soc. 8 _(1933), 254-260.

[CE] A. Copeland, P. _Erdös, Note on normal numbers. Bull. Amer. Math. Soc. 52 _(1946), 857-860.

[Cs] I. Csiszar, T. M. _Coven, B.-S. _Choi, Conditional limit theorem under Markov conditioning.

IEEE Trans. Inform. _Theory IT-33 6 _(1987), 788-801.

[DK] M. _Denker, K. F. _{Krämer, Upper} and lower class results _{for subsequences of} the

Champer-nowne number. _{Ergodic theory} and related _{topics III,} LNM 1541 _{Springer (1992),} 83-89.

[E1] R.S. _{Ellis, Entropy, large} deviations and statistical mechanics. _{Springer-Verlag,} 1985.

[Go] I. J. Good, Normal Recurring Decimals. J. Lond. Math. Soc. 21 _(1946), 167-169.

[Ma] J.E. Maxfield, Normal k-tuples. Pacif. J. Math. 3 _(1957), 189-196.

[NS] Y.-N. Nakai&#x26;I. Shiokawa, Discrepancy estimates for a class _of normal numbers. Acta Arith. 62 _(1992), 271-284.

[Tu] R. Tutte, Graph Theory. Encyclopedia of Mathematics and its Applications Vol 21 Addison

Wesley, 1984.

Edgardo UGALDE

IICO-UASLP,

Av. Karakorum 1470, Lomas 4ta

San Luis Potosi S.L.P., 78210 MEXICO