Sets of block structure and discrepancy estimates

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

R

EINHARD

W

INKLER

Sets of block structure and discrepancy estimates

Journal de Théorie des Nombres de Bordeaux, tome

9, n

o

_{2 (1997),}

p. 337-349

<http://www.numdam.org/item?id=JTNB_1997__9_2_337_0>

© Université Bordeaux 1, 1997, tous droits réservés.

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

Sets of

block

structure and

_discrepancy

estimates

par REINHARD WINKLER

RÉSUMÉ. Soient x =

(xn)n~N

une suite d’éléments d’un ensemble

fini M et f =

(fn)n~N

une suite

_{d’applications}

_{fn :}

M ~ M.

Quelle

information sur x et f _permet d’obtenir des estimations de

la

_discrépance

de la suite

_{f(x) = (fn(xn))n~N?}

Nous donnons dans cet article des

_réponses

à cette

_question,

en utilisant un

résultat

_qualitatif

récent.

ABSTRACT. Given a _sequence x =

(xn)n~N

on the finite set M

and a _sequence f =

(fn)n~N

of _maps

fn :

M ~ M. Which

information about x and f is suitable for

getting

estimates for the

discrepancy

of the sequence

f(x) = (fn (xn))n~N?

The

paper’s

object is, using

a recent

qualitative result,

to

_give

answers to this

question.

1. Introduction

Let M be a finite

_set,

_w.l.o.g.

M =

{1, ... , m~.

For a _sequence x =

on M and a _sequence f = of

transformations In :

M -~ M

we are

going

to

study

the distribution behaviour of the _sequence

f(x) =

y = =

fn(xn).

The

qualitative question

is: Which f have the

property

that _y is

_uniformly

distributed

_(u.d.)

whenever x is u.d.?

(As

general

references for the

_theory

of u.d. _sequences cf.

_{(Kui-N), [H]}

or

(D-T~.)

An answer is contained in

[W]

_by

means of a characterization of all those

so-called

_u.d.p.

_(uniform

distribution

_preserving)

f.

_Proposition

2 recalls the result

_adopted

for our _purposes. In this _paper we show in which _way one can derive

quantitative

versions which are _necessary for

applications

similar to Monte Carlo and

_Quasi-Monte

Carlo methods. More

_precise:

We

are

looking

for functions

FN

such that

holds.

_DN(x)

denotes the

_discrepancy

_which,

for an

_arbitrary

_sequence x on a finite set M of

_{cardinality #M}

= _m, is defined in the

following

way.

For

_arbitrary

T C N

_put

and

_(for

_{N &#x3E;}

min

_T,

_{T 0}

₀₎

If T = N _we

shortly

write

A(N, x, a)

resp.

DN(x)

for the

discrepancy.

Drr(x)

tends to 0 for N -~ oo if and

only

if x is u.d.

Similarly

s-block

discrepancy

is defined

_by

= where

is considered as a _sequence on M~

_consisting

of the members

A _sequence x is called s-block u.d. if

D~r,~(x) -&#x3E;

_0,

x is called

completely

u.d. if this holds for all s

Section 2

_presents

Theorem 1 which shows

_why

_FN

has to

_depend

on x in a more

complicated

_way than

_just

via

DN(x).

_Interesting

and much

deeper

results on similar

complexity questions

on uniform distribution can

be found in

_[G]

and

_[Ki-Li].

Section 3 is devoted to the notion of sets of block

structure which has been introduced in

_[W]

and is useful for our _purposes.

This notion is

_closely

related to the

_concept

of almost constant _sequences

which has been studied

_by

Rindler and

_Losert,

cf. for instance

_[Ri],

[Ri-Lo]

and

_[Lo].

_They

continued

_{investigations}

due to

_Rauzy,

cf.

_[Rau],

who

seems to be the first one who

_investigated

_questions

on

u.d.p.

_sequences in

a similar _way. For more references cf.

~W~.

In this _paper

(section 3)

several

simple properties

of sets of block structure are

_proved

which are relevant

for us. Section 4

_analyses

how the

_qualitative

characterization of

_u.d.p.

f in

Proposition

2 can be modified in such a _way that it can be used for concrete

and

_quantitative

_discrepancy

estimates. The discussion is

_essentially

lead

by

the results of Section 2 and 3. A

_discrepancy

estimate of the desired

type

is carried out in the final section 5. The

_problem

is discussed also for s-block

_discrepancy.

One

easily

gets

a characterization of those

u.d.p.

f

which also preserve s-block or

complete

uniform distribution.

2.

_Discrepancy

alone is not

_enough

information

Our main

_question

is: In which _way should

_Fnr

_depend

on its

arguments?

If

_FN

is allowed to be an

arbitrary

function the

setting

of the

question

is too

general,

since

_{FN(x, f)}

=

DN(f(x)),

in a trivial _way, is the

optimal

solution

but not

_interesting

for

_{computations.}

The first restriction one thinks of is to

_require

that

FN

depends

in the first

_argument

not on all information

about x but

_just

on the

_discrepancy:

_FN

=

FN(DN(x), f).

But this cannot be done in a reasonable _way as Theorem 1

explains.

THEOREM 1. There is no

family of functions

_FN

=

FN(DN(X), f),

N E

N,

each

_FN

_depending

on x

only

via

DN(x)

and on f =

(In)nEN,

fn :

M -

M,

in an

_arbitrary

_way with the

_following

_properties:

for

all N and

whenever x is

uniforml y

distributed and f _preserves

uniform

distribution.

Proof:

_{Suppose, by contradiction,}

that such a _sequence of

FN

exists. For

technical convenience suppose M =

(0, 1)

and define g : M - ~VI

by

g(i)~i,i=0,1.

Let

_{fn = IdM if aj}

j =

0,1,

... , where

the _sequence

is defined in such a _way that aj =

_3aj-l

and

_2aj

for j

&#x3E; 2. Then

f = is

u.d.p.

This follows from the main result in

_[W],

here restated

as

_Proposition

2 in section 4. Consider the sequence x = defined

by xn

= i E M with n - i mod 2. x is u.d.

and, by assumption,

Now define _y =

(Y,,),,Elq

in such _{a way}

that yn

= 1 if

In

=

IdM

and yn

= 0 if

_fn

= _g. Hence

In(Yn)

= 1 for all n _E N and

DN(f(y))

= 2

for all N E N. Note

_Da;(Y)

=

Da;(x)

= 0 for

all j

&#x3E; 1. For

arbitrary -

_&#x3E; 0 and

sufficiently

large j

this

_yields

contradiction.

_q.e.d.

Thus one has to make finer

_{investigations}

of u.d. _sequences x and use

properties

which fit to

_u.d.p.

_sequences f in the sense of

_Proposition

2

(cf.

section

_4).

For that we have to turn to the

_concept

of sets with block

structure.

3. Sets of block structure and related

_concepts

For a better

_{understanding}

of our

problem

we consider

partitions

of the set N of

_positive

_integers

into blocks. For this reason we use the

following

notations. If

0

is a

_given

_sequence of

_integers

let

_Ik

=

(ak-1,

ak]

n N denote the induced blocks

_(intervals)

and call the

_family

I =

(Ik )kEN

the

partition

of N induced

by

a = If 0

(or,

equivalently,

ak+llak

-

1)

for h --~ o0 we call I = short

partition.

We introduce the abbreviation

where

_k(N)

is the

_greatest

index such that

_ak(N)

N. If this

_quantity

is small this means that almost each block

_Ik

is almost contained either in T or in its

complement

Hence it is obvious

why

we _say that T has block

structure if

_L(T,

_I,

_N)

= 0 for all short

partitions

I. It is easy _to

see that this

_is,

for

_instance,

the case if T is

_compatible

with a _sequence a = with _&#x3E; 1

(example:

a (q) =

q &#x3E;

_1),

in

the

_following

sense: A set T C N is called

compatible

with a _sequence a

(or

with the induced

_partition

I =

I(a) = (Ik)ken)

if and

only

if each

Ik

is contained either in T or in its

complement

~1 ~

T . With ,~ we denote the

system of all sets T C N with block structure. Sets of block structure have been characterized in

_[W]

in several _ways.

A related

_topic

which is

_important

for us is the distribution of restricted

sequences. For TON define

and for xT =

1,

call the

positive

number

the restricted

_discrepancy.

We _say that XT is u.d. w.r.t. T if

DN(xT)

-0 for N -- oo. Note that

DN(x, T )

where,

in

general,

the

inequality

is strict.

The _sequence of numbers

_densN(T)

E

_~0,1~,

N E

N,

has an _upper limit

dens(T),

called _upper

_density

of T.

_Similarly

the lower

_density

_dens(T)

is defined as the lower limit. In case of coincidence of both the common value

dens(T)

is called

_density

of T. It is _easy to check

_that,

for

_arbitrary

sets

S, T

C N of natural

_numbers,

_{d(S, T)}

=

W(S4lT)

defines

a pseudometric

d

_inducing

a

topology

on the _power set of N. Here =

(S ~ T)

_U

(T B

S)

denotes the

_symmetric

difference of the involved sets. Note that the

topological

closure of the

_singleton

_{0},

for

_instance,

contains

_exactly

the

sets T with

_dens(T)

= 0.

Here are

descriptions

from

[W]

of sets with block structure:

PROPOSITION 1. For T C N the

_following

three conditions are

_equivalent:

(1)

T E B.

(2)

For all - &#x3E; 0 there is _{a q} &#x3E; 1 and an S

compatible

with

a (q)

such

that

_{d(S, T)}

E.

(3)

For all u.d. _sequences x on a

_finite

set M with

_{#M &#x3E;}

2 the

restric-tion XT is u, d. zv. r. t. T.

Proof: Follows from

_[W],

Theorem 1.

_q.e.d.

The

_following

theorem

_presents

further useful observations on B.

THEOREM 2. B is a Boolean set

_algebra

_(not

a

a-algebra)

on

_N,

and,

as a

subset

_{of the}

_power set

_{of N, it is topologically}

closed w.r.t. d.

Proof: ,Ci is not a

Q-algebra

_since,

for

_instance,

_An

=

{2n}

_E

l3,

but A =

UnEF1

An

B.

_Obviously

,CB contains

_0,

N

_{and, immediately by definition,}

is

closed under

_complements,

i.e. T

_implies

_{N B}

T E Ci. To show that B

is a Boolean set

_algebra

on N it therefore suffices to _prove that S U T E ,Ci

under the

_assumption

that

_S,

T E B: For _any

_given

short

_partition

I define

sk + tk . This

_implies

since ,S‘ and T are

_supposed

to have block structure. Thus ~5’ U T E B.

Finally Proposition

1.2 says ~3 = C for

hence Ci is indeed a closed set.

_q.e.d.

From the results _up to now it follows

that,

for T E ? and E &#x3E;

_0,

there is

a set S E Ci

_compatible

with some a = with

ak+l~ak &#x3E; q

for some

q &#x3E; 1 such that

_{d(S, T)}

E. Since d is a

pseudometric

and not a metric

the

_question

arises whether one can even take 6’ = 0. That this is _not the

case follows from Theorem 3.

THEOREM 3. There are sets T with block structure such that

_d(S,T)

&#x3E; 0

for

all S

_compatible

with some I =

I(a),

_a =

(ak)kEN

with _{q &#x3E;} 1

or,

equivalently,

Proof: Construction of T: For

_j,

I &#x3E; 1 define the numbers

_{r( j, l)}

=

2 2j+l

₊

2’,

=

22j+l

+ 2 ~ 21 =

22j +1

₊

2’+’

and the _sets

We claim that the set T =

Too =

Aj

has the desired

_properties.

_(Note

that n E

_Ai

if and

_only

if the

_{binary representation}

of n - 1 starts with a

block 100 ...001 where the number of

_0-digits

between the first two

_1-digits

is

_{2j - 1.)}

T E B: The set

_Aj

_is,

_{by definition, compatible}

with the _sequence

Check that the

_quotients

_{of succeding}

members of this sequence are bounded

below

_by

1 ₊

_{(2 2j}

+

_1)-1

_{= qj} &#x3E; 1.

_{Hence Aj}

for

_{all j}

and

_thus,

_by

Theorem for all i E N. Now observe

J=Z

for

_{sufficiently large i.}

Thus T E

_N}

C .C~ = .f3

by

Theorem 2. It is obvious that the both

_remaining

assertions of the theorem are indeed

equivalent.

We

_prefer

to _prove the second one. For that reason _suppose now that I =

I(a),

a = is a

_partition

with _q for some q &#x3E; 1. If the

_quotients

are not bounded it is easy to see that

Hence we _{may suppose}

Q

for some fixed

Q

to show that

_L(T,

_I,

_N)

has a

_{positive upper}

limit.

For each n let

kn

be chosen in such a _way that For

q = such that

2-2~

~/3.

2j

+ 2 and I = n -

_2j

we

distinguish

two cases:

First case:

_ak"

With the notation

and,

using

From the definition of T follows

Second case:

r( j, l)

_{akn. Similar} to the first _case,

_{just replacing j by}

j

+

1,

1

_by

1 - 2

and kn

by

one

_gets

Combining

both cases and

_{choosing E}

=

22Ji2Q)

one

_gets

Theorem 3 and the

_following

fact will also be relevant on the search for

possible

estimates,

cf. the discussion in section 4.

THEOREM 4. Let

_{(I(’))’EN,}

1(1)

=

I(a(l)) =

a(l) =

be an

arbitrary

countable

_{family of}

short

_partitions.

Then there is a set T

_ç

N with

_{L(T, I~i~, ~ ) -~}

0

_for

atI l E l~ but

_{T ~}

B.

Proof: Since all are short

_partitions

there is a _sequence

such that

#

₁

whenever I d and and with

_{N1+1/N1 --1-}

oo.

la _ k _ t

1+1’

i

k

Furthermore there exists a short

_partition

J = =

J(b), b

= such that

_{sufficintly large}

I

_{fulfill t2}

if

_{Jk n}

0.

Define

..-- i 1 .

To estimate

note that

_only

those k contribute a

notvanishing

term for which

_{bk, E}

Iki)

for some k’.

Using

the bounds for and

~Jk/bk

we conclude

On the other hand let I =

I(a)

be _a short

_partition

with )

if

Ik

n

_(Ni,

_{Ni+i ) # @}

for

_{sufficiently large}

1. Since

_#-~I~’-

- 0 for

and

_{k, k’}

2013~ 00 a further use of the bounds for and those for

yields

implying

lim

_L(T,

_I,

_{~V) &#x3E;}

_3.

Thus T does not have block structure.

q.e.d.

4.

_{Reconsidering}

a

qualitative

result

For

_given

f = ~VI --~

_M,

_we consider the _sets

T~ _ ~ n

_E

I

fn

=

7r}

for _every

_permutation

_{7r E}

SM

of the _set M

(SM

denotes the

symmetric group

acting

on the set C =

~n

_E

N I In

=

c,~~

of all n

for

_{which fn}

is a constant function

_taking

some value cn and the

remaining

set D =

N B

C B

T where T =

T1I".

Now we are

ready

to formulate the characterization of

_u.d.p.

_sequences f = of _maps which is the motivation for our

quantitative

results.

PROPOSITION 2. The _sequence f =

f,~ : M -&#x3E;

M,

is

u.d.p. if

and

only if

the

_following

three conditions hold:

1.

_dens(D)

= 0.

2. All sets

_{C, D}

and

_T1r,

7r E

_SM,

have block structure.

3. The _sequence cc =

(Cn)nEC

is u.d. _w.r.t. C.

Proof:

_{Follows, by specialization, immediately}

from

_[W],

Theorem 3.

q.e.d.

If we _say that f is

compatible

modulo constant _maps with a

_{partition P}

of N if on each T either all

_In

coincide or all

In

are constant _maps then we can reformulate condition 2 of

_Proposition

2 as follows: f is

_compatible

modulo constant _maps with a finite

_partition

of N into sets with block

structure.

In order to use the conditions from

_Proposition

2 for

discrepancy

esti-mates we have to

_analyse

how

_they

can be made concrete

_by

means of

suit-able

_quantities.

For condition 1 and condition 3 this is clear:

_dens(D)

= 0 if and

_only

if the

_quantity

_densN(D)

tends to 0 and c = is u.d.

w.r.t. C if and

_only

if the

_quantity

_DN(CC)

tends to 0.

Which

_quantity

describes how

_good

the block structure of some T 6 ~

is? This

_question

is not so trivial. The reason is that in the definition for

T the limit relation for

_L(T,

_I,

_N)

must hold for

_all,

even

uncountably

many, short

partitions

I.

Unfortunately

the behaviour of one fixed I or

even of

_countably

_many of them does not

_give

_enough

information. This has been illustrated

_by

Theorem 4.

Looking

at certain

_{special examples}

of _sequences with block structure

(as

the sequences

a (q)

for q &#x3E;

_1),

one observes that

L(T,

I,

N) -

0 for some

suitably

chosen

_"long"

_partition

I with &#x3E; 1. The

_hope

that this

_might

_happen

for all T has been

_{destroyed by}

Theorem 3. But

what we _may

_expect

is that

L(T,

I,

N)

_gets

small in

dependence

on N and

b if 1 + 6-

_(Let

as call

_partitions

I =

I(a)

with this

_property

6-partitions.)

This

_approach

indeed is

_possible

as Theorem 5.1 will show.

Theorem 5.2 is an

application

of Theorem 5.1 to

u.d.p.

f.

But how to match a u.d. x with a

_u.d.p.

f? Condition 2 in

_Proposition

2 tells us that in a

u.d.p.

f the

_permutations

7r : M --~ M occur

(with

few

exceptions)

in

_long

blocks. On such blocks the distribution behaviour of the _xn is the same as the behaviour of the

7r(xn) (up

to a fixed

permutation

7r of the elements of

M).

Hence we have to

_guarantee,

that the blocks are

_{long enough}

such that the uniform distribution of x can be

observed. This will be the

_object

of Theorem 5.3.

THEOREM 5.

_{( 1 )}

_Suppose

that T has block structure and &#x3E; 0 is

_given.

Then there exist a

_positive

b =

6T(E)

_&#x3E; 0 and a natural number

such that

1 11 -, 1 /

holds

_for

all N &#x3E;

_NT(£,6)

and all

_b-partitions

I =

(1k)kEIq-(2)

Suppose

that E &#x3E; 0 and f

_u.d.p.

are

_given.

Then there is a

_positive

b =

~f(~)

_&#x3E; 0 and a natural number

N1=

6)

such that

for

all

N &#x3E;

_N1

and all

_6-partitions

I there is an f’ =

compatible

with I modulo constant _maps such that

Furthermore we _may assume

DN(f(x)c)

-

_for

such N.

(3)

Suppose

that x is u.d. and _E, b &#x3E; 0 are

given.

Then there is a

positive integer No

=

b)

_E ~I such that

for

all I =

(N’,

Ni]

n N with N’ &#x3E;

_No

and

₍₁

+

_6)N’

and all N &#x3E;

_N1

Proof:

(1)

Suppose,

by contradiction,

that there is some E &#x3E; 0 such that for ah 6 &#x3E; 0 and all N there is a

6-partition

1(6, N) = (Ik(6, N))keN

By

a standard

_diagonal

_argument

_(similar

as in the

proof

of The-orem

4)

the

partitions

1(/)

=

I( t, N1)

with

sufhciently rapidly

in-creasing

Ni

&#x3E; and

_{corresponding Ni}

can be combined to a

short

_partition

I with

implying

contradiction.

(2)

1

x E

_?M}

where

_{C, T7r}

~ 8 and

_{D ~}

S are as

above. All S

_{E S, by Proposition}

_2,

have block structure. Thus

_part

1 of the theorem

_guarantees

that there _are, for each S E S and for

2~+2) ?

numbers and with the

_properties

stated there. Let b =

6f(s)

_&#x3E; 0 be the minimum of the

s E S,

w.l.o.g.

E/2.

Furthermore, by Proposition

2,

there

is a natural number N’ =

ND(£1,8D(£1)),

such that

densN(D)

_£1 for all N &#x3E; lV’, Define

Now choose _any

_6-partition

I =

I(a).

We have _to find _an f’ = with the

_properties

stated in the theorem. For all n E

11

let

_{f ~}

be the same

arbitrary

but fixed

mapping.

For n E

Ik,

k &#x3E; 2

distinguish

two cases:

°

Case 1: If

_{E Ik ~}

_fn

=

7r}

_&#x3E;

#Ik/2

for _{some 7r E}

SM

_put

for all n ~

Ik.

Case 2: Otherwise define

_f,~

=

_cn

for some constant

_cn

where

_cn

= _Cn for the case

_{that f n =}

_c,~ is constant.

By

definition f’ is

_compatible

with I modulo constant _maps. We have to estimate

_densN(X)

for X =

~n

_E

N

I

fn}

and N &#x3E;

Nr(e,6).

Let

_{Ak =}

_and,

for S E

_S,

_{AS,k = min(#Ik n}

s, ~Ik ~ S).

In case 1 we have

Ak -

_ATr,k

for some ~r E

_sM.

In

case 2 we have

Ak

=

#(I~ 1 C)

and _we

distinguish

_two subcases: If

C) &#x3E; ~(Ik 1

C)

then

_Ak

=

Ac,k .

In the other subcase _we

get

lY ’ I

which is the assertion of the second

_part

of the theorem.

(3)

Follows with the same method as Lemma 1 in

[W].

q.e.d.

5.

_Discrepancy

estimates and s-blocks

Using

Theorem 5 we are now in the

_position

to deduce a

discrepancy

estimate for

_f(x)

of the

_following

_type:

THEOREM 6. Given a u.d. _sequence x and a

_u.d.p.

_sequence f

_of

_maps.

For

_{arbirarily given}

e &#x3E; 0 let b =

bf(s)

1~

and

Nl

=

Nf(E,

6)

(w.l.o.g.

N1

2::

_a)

be as in Theorem 5.2. Furthermore let

80

E

_{(0, b -}

2

and

_No

N’

N1

where

No

= as in Theorem 5.3. Then

for

all N &#x3E;

_Nl

the

_{discrepancy of}

the _sequence

_f(x)

can be estimated

by

Proof: Fix _any

_partition

I =

I(a)

with 1 ₊ 1

-E- b

and

~Nl/2

EN,.

_(Note

that this is

_{possible by}

the

_assumption

on

bo

since

_1.)

Let f’ be as in Theorem 5.2

_compatible

with I modulo

constant _maps and let the sets _resp.

_C~D~T~T~

be associated

to f _resp. f’ as in section 4. Define for

arbitrary

N

and the numbers

_I~o

and

_1~1

_by

N’ _{ako resp. 0~1} N _{ak1 +1.}

Combination of these estimates

_gives

Since E and 6 can be chosen

arbitrarily small,

this result

gives

a method to determine how

_large

N must be such that

_DN(f(x))

_gets

smaller than

any

given

positive

value. For

given

u.d. x and

u.d.p.

f one has to know the

integer

functions

_Nx(£,6)

and

_Nf(ê,6)

which exist

_by

Theorem 5.

It is clear that the

_explicit

bounds for

_DN(f(x))

_stemming

from the

proofs

of Theorem 5 and Theorem 6 are not

_optimal.

But a more careful

estimation would have taken so much technical effort that the view to the main

_object

would have been obscured

_by

_boring

calculations. What we

want to

_emphasize

is that functions

_Nx(s)

and

_Nf(s)

_{depending only}

on one

argument E

do not

_{give enough}

information to

_guarantee

_{DN(f (x))}

_6~ but the functions

_{N x( ê, b)}

and

₆₎

_{guaranteed by}

Theorem 5 and used in

Theorem 6 do.

Finally

we have to

_investigate

how our methods

apply

to s-block

discrep-ancy. If one assumes that f is

u.d.p.

then the

generalizations

are

_straight

forward. The

_only

modifications are as follows:

In

_estimating

one has to look at the blocks

If

_only

one of

the Xi

or fz, i = n, ... , n + s -1,

does not behave

regularly,

this _may affect not

_only

one but s members in the _sequence

yes) =

Hence

_discrepancy

estimates have to take this into account

_by

a factor s.

It is clear

_that,

for _every fixed s E

N,

the

_asymptotic

definition of block

structure causes that this

_problem

does not affect the nature of the results. Of course we have to

_modify

the definition of restricted

_discrepancy

in the

following

way: For T C N and a E MS define

-- ----

I ... -

I

x is called s-block u.d. w.r.t. T if

DN(x) -

0 for N - oo.

Similarly

one

calls x

_completely

u.d. w.r.t. T if this is the case for all s E N.

In the natural _way we call f s-block _resp.

complete u.d.p.

if f (x)

is s-block

resp.

completely

u.d. whenever x is s-block _resp.

completely

u.d. With the same notations as in

_Proposition

2 the

_qualitative

result is the

_following:

THEOREM 7. Let f be

_u.d.p.

Then f is s-block resp.

completely u.d.p. if

Proof:

_Carry

out the _program described above. We omit details.

_q.e.d.

Note that in Theorem 7 we have assumed that f is

_u.d.p.

It is a

conjecture

but

_yet

_unproved

whether s-block or

_{complete u.d.p. always implies u.d.p.}

REFERENCES

[D-T]

M. Drmota and R. F.

_{Tichy. Sequences,}

_discrepancy

and

_{applications.}

Springer,

Lecture Notes in Mathematics 1651

_(1997).

[G]

M. Goldstern. The

_{complexity of}

_uniform

distribution. Math. Slovaca 44.5

(1994), 491-500.

[H]

E. Hlawka. Theorie der

_{Gleichverteilung.}

Bibl. Inst. Mannheim Wien

-Zürich,

1979.

[Ki-Li]

H. Ki and T. Linton. Normal numbers and subsets

_{of N}

with

_given

densi-ties. Fund. Math. 144

_(1994),

163-179.

[Kui-N]

L.

_Kuipers

and H. Niederreiter. Uniform distribution of _sequences. John

Wiley

and

_Sons,

New

_York,

1974.

[Lo]

V. Losert. Almost _{constant sequences}

_{of transformations.}

Mh. Math. 85

(1978), 105-113.

[Lo-Ri]

V. Losert and H. Rindler. Almost constant _sequences. Soc. Math. de

France

_Astérisque

61

_(1979),

133-143.

[Rau]

G.

_Rauzy.

Etude de

_quelques

ensembles de

_fonctions

_définis

_par des

pro-prietés

de _moyenne. Théorie des Nombres Univ. de Bordeaux

_1972/73,

Exp.

20.

[Ri]

H. Rindler. Fast konstante

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Acta Arith. 35.2

_(1979),

189-193.

[W]

R. Winkler. Distribution

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sequences

of

maps and almost

con-stant sequences on

finite

sets. To _appear in Mh. Math.

Reinhard WINKLER

Institut fur

_Algebra

und Diskrete Mathematik TU Wien

Wiedner

_HaupstraBe

8-10

A-1040 Wien Austria