On the diaphony of one class of one dimensional sequences

VOL. 19 NO. (1996) 115-124

ON

THE DIAPHONY

OF ONE

CLASS

OF

ONE-DIMENSIONAL

SEQUENCES

VASSIL ST. GROZDANOV DepartmentofMathemancs

UniversityofBlagoevgrad

2700Blagoevgrad, Bulgaria

(ReceivedNovember7, 1991 and in revisedform

August

15, 1992)

ABSTRACT In the present paper, we consideraproblemof distribution ofsequences in the interval

[0,1),

the so-called ’Pr-sequences’ We obtain the best possible order

O(N-(logN) /’2)

for the diaphony of such

Pr-sequences

For the symmetric sequences obtained by symmetrization of

P-sequences,wegetalso thebest possibleorder

O(N-l(lo9

N)

1/.2)

of the quadratic discrepancy

KEYWORDSANDPHRASES Distributionofsequences, quadratic discrepancy andPr-sequences 1991AMS SUBJECT CLASSIFICATIONCODE 11B83

INTRODUCTION

Leta

(x,),:0

be an infinitesequenceinthe unit intervalE

[0,

1)

For everyreal number

xEE and every positive integerN we denote

AN(a, x)

the number ofterms x., 0

<

n

<

N-

1.

whichareless thanx

Thesequence iscalleduniformlydistributed inEiffor everyrealnumberxEEwehave limN_,AN(cr; x)N-1

x.

Thesystematic studyof thetheoryofuniformlydistributedsequenceswasinitiatedby Weyl A classical measure for the irregularity ofthe distribution of a sequence a in E is its quadratic discrepancy

TN

(or),

which is definedfor every positive integerNas

TN(O’)

(f

AN(a; x)/N-

x

12dx)

1/"

The irregularity ofdistributionwithrespect to the quadratic discrepancywasfirst studiedby Roth [2]

In 1976, Zinterhof(see [3,4])proposeda new measure fordistribution, whichhenamed diaphony Thediaphony

FN(a)

ofais definedforevery positive integerNas

2 h-9

N-1

(a;h)

FN(O-)

-’]h=l

SN

/2

where

SN(O

h)

’=0

N-1 exp(27rihx,) signify trigonometricsumofa

Wenotethatthe diaphony ofacanbewritten inthe form

FN(O-)

(N

-

N-1 where

g(x)

--7v2 (Zx

2x

+

1/3)

Itiswell known

(see

[5],p 115,

[4])

that both equalities

2 Using thewell-known theorem ofRoth

[2]

it can be proved (see Neiderreiter [7], p 158; Proinov

[8])

that for anyinfinitesequenceainE,theestimate

TN(a)

>

214-N-(logN)

/’2

(1.1)

holds for infinitely many integers N The exactness ofthe order of magnitude ofthis estimate was proved byProinov([9],[10], [11])

Proinov[8] provedthat for any sequenceainEthe estimate

FN(Cr)

>

68-1N-l(logN)

1/2 _(1.2)

holds for infinitely manyN.

From (1.1) and (1.2) becomes clearly that the best possible order of diaphony and quadratic discrepancyofevery sequenceain

E

is

O(N-

(logN)

1/2

).

2. A SEQUENCE OF r-ADIC RATIONAL TYPE.

2.1 CONSTRUCTION OF SEQUENCE OF r-ADIC RATIONAL TYPE

Inthispartwegeneralize Sobol’s([12], [5],p 117, [13], p. 23)constructionofsequencesof binary rational type

Letr

>_

2 isfixedinteger. Weconsiderthe infinitematrix

(vs,3)

vm

v

(2.1)

where for every s, j 1,2,...,v_s,3

{0,1,..-,

r-

1}.

We suppose that in every column, the quantityofvs,j, which aredifferentfromzerois apositiveintegernumber,i.e.,

vs,

0forjsufficiently

big Suchmatrixweshallcall guidingmatrix

Toevery column of thematrix(2 1)correspondsar-adic rationalnumbers

V

0,

v,

v,.

.v,j.

(s

1, 2.

(2.2)

The numbersdetermined in(2.2)arecalled guiding numbers. We signify

No

NU

{0),

withNthesetof natural integers.

A sequenceof r-adicrationaltype(or RP-sequence)is asequence

((i)),:0,

which isgenerated by theguidingmatrix

(v,j)

inthe following way: Ifinthe r-adic number system

i-8mem_ 1

thenin the r-adicnumber system where for j 1,2,

,

m

Wj

eV

VfVf

"V,

(2.3)

e

terms

and isthe operationofthe digit-by-digit addition modulorofelements of

Z

{0,

1,

,

r

1}.

A

RP-sequence

(o(i)),=0,

which isgenerated bytheguidingmatrix

(v,l)

canbe alsoconstruct:l by followingthe threementionedbelow rules

()

(0)

O.

(2) If/=

r(z e No),

thenqo(i)

Vs+l.

(3)

Ifr

<

<

r

s+l,

then

o(i)

e+

(r)*(i-

e+ r),

where

es+

ishigher significant digit in r-adicdevelopment of and

e+ (19(

rs

ys*+

1Vi*+

1"

"*

Vs+

1.

es+

terrr8

Obviously the operation hascommutativeandassociativeproperty.

We

shall prove thatthetwodefinitionsof the

PR-sequenes

areequivalent. Letussupposethat the firstdefinition is validfor

RP-sequence.

(1)

If 0, thenobviously

(i)

0.

(3) Let us assume that r <

<

r+1 and

(%+1%-..e)r

Since the operator is commutative andassociative we have

(i)

O,

((1 V1)*"

"(ca

V))*(%+1

Vs+I).

Since

Vs+l

(r

)

and %+

(es

e

el)r

then

(i-).

Finally

(i)=

%+l(r’)’(i

e,+

).

The three roles in the second definition for

RP-sequenceeproved.

Reversely,let the second deflation for

PR-sequence

is validd isgiven positive integer. Then there

ests

uquely positive integersthatr

S <

r

+1.

We shall provedeflation byinduction ons. Ifs 0,thenI

<

rand

(i)

iv()’(0)

0,ivy.

Wemakeinductivesupposition that forsomes Nand

eve

integer i,r-1

S

<

r deflation holds. Letus assumethatr

S

<

r+a and

(e+%

el)

Fromrole3 wehave

If we denote j=i-%+l r

,

then j=

(e e_...el)

d

r-l

j

<

r

.

Then by inductive supposition

(j) 0,

(elY,)"

"(e,).

By

role 2,

(r

)

+1

dwehave

(i)

0,

(el)*

*(%)*(%++).

Deflation holds for

eve

positiveinteger

.

In

thefollonglena we

ve

a

prope

of the nctions

.

LE

2.1.

Let

(vo)

isan

bitr

iding matrix, d

((i)),0

is

RP-sequenee,

wehis generated by

(v,s).

Let v, m, nbe integer numberssuchthatv

No,

0 n

<

r dm

O(mod

r").

Thenwehave

(

+

)

()’().

Theproofof the lenais obvious.

For

eve

integera

Z

wedefine the

oy

integer, wehis a lutionof the equation

a

+

O(mod r).

If =0,12. "or, where, for r=1,2,. .,t

oZ,

then we dee 0,12-"

"t.

2.2.

SEQNCES

OF r-IC TION

TE,

CH P-SEQNCES.

The

theo

of the

P-mquenees

was first studied by Faure

([14];[15])

d

gener

by Yeideneiter

([ 16];[17]).

Ar-adie

element

inte is inte

t,

[(-

1)/,j/),

inweh I

S

j

S

r

,

foryimegerm. Let N r

.

We shl 1 thenet

X

(z0,

z,

.,

zu_)

be anetof

e

P

(or

P-e),

if

eve

r-adie

element

inte

l,s,

hating

lenh

1/N

ntn

onepoint ofthenet

X.

A

r-adie section of the sequence

X

(z),0

isa set oftesz,, th numbers i, mtisng the inequities

kr

< (

+

1)r

,

for

eve

integers k and

,

such that k 0, 1, 1, 2,

Thesequence

(z,)io

iseledasequenceof

te

P

(or P-suenee)

if

eve

r-adie sectionis a

P-net.

I W21 U31 U71

0 1 V32 V./2

(v,.3)

0 0 1 v33

0 0 0 1

Then the corresponding

RP-sequence

is

Pr-sequence

PROOF. We choose arbitraryr-adic sectionofthe

RP-sequence

(o(i)),0,

thelength ofwhichis r’. Wewritethe numbers i,belongingtothis section inthe r-adic number system:

cucu_

"cm+lemem-1 "el, (2 4)

whereckarefixedandekarearbitraryr-adicnumbers

Wechoose now anarbitraryr-adic intervall, withlength r-’- Inther-adic system this interval is determinedbythe inequality

O a a2 a,

<

x

<

O a a2 "am+0, 1,

rn-zero8 whereal,

,

a,arer-adicnumbers

We shall prove, thatfor everychoiceof the numbersckand akamong the numbersi, in theform

(2.4)

thereexistsexactlyonei, forwhich

o(i)

E l. Inther-adicnumber systemwewrite

y)(i)

O,g,,1L.2

g,.3 From (2.3)wehave

g,,3

elV,3

ernVn,3Cm+lVn+l,3

where the sense

ofekvk,

isthe sameasin

(2.3).

Thecondition

o(i)

E isequivalenttothe following conditions

g,o=a,

forl_<j_<m. Weget that foreach j, 1

<

j

<

rn

g,../

(elVlo.

emVm,3)

(Crn+lVm+l,3"

.*C#V#,3)

fromwhich weget

ev’,,.

*e=vm,

aj(cm+v:.+,,

....

c.v.,,)

(1

<

j

<

m)

Letuscall

f

thefight-sideof

(2.5)

for1

_<

j

_<

m. Havingin mindthat fors 1, 2,

,

vs,s

1and in case j

>

s,

v,,

0, the system

(2.5)

become

ev;oe3+lV+l,...

*emv,,

L(1

_<

j

< m).

In

this system the unknowns el,e2, ",

ern

aresuccessively so determined thatit has onlyone

solution.

The theorem isproved.

Inthefollowinglemmaweshallshowsomeproperty of

Pr-sequences.

LEMMA

2.2. Let N

r"

where u E

No.

For every guiding matrix

(v,o)

in which

v,,,

1and

v,,

0 for j

>

s(s

1, 2, and for the

RP-sequence

(o(i)),__0,

which isproduct of

(v,,j)

we have

{o(i):

0

<

<

r"}

{fiN:

0

_<

j

<

N}

(2.6)

PROOF. Weshall make theproof byinduction on u. Ifu 0andu 1, thenwemake directly examination.

We make inductive supposition, that for some u N the equality

(2 6)

is true and for j 0,1, .,r-1 we considerthe multitudes

A3

{o(i)’jr

<

<

(j

+

1)r"}.

Then obviously

A

U:

A

(2.7)

where

A

{o(i):

0

_<

<

r+l}.

Weconsiderthat j 0.

By

the inductivesupposition

Letus now considerthat1

<

j<r 1. Weshallprove the following equality

A

{m/r

"+L₀

<

_m

<

_r

+’,

_m=__j

(rood

r)}.

_(2.9)

Letj, 1

<

j

<

r 1is fixedinteger andconsiderthat jr"

<

<

(j

+

1)r

". Letusrepresent inthe formi=jr"+k,whereO<_ k<r

.

Thenby Lemma2 wehave Itisobviousthat

Letusput

)(i)

(jr")*o(k)

(2.10)

(j

rv)

Y;+l*+l

*Vv+l

(2.11)

j-

erms

(jr

v)

0,Wv+I,lWv+I,2" "Wv+I,v+I.

From (2 l)isclear,thatwv+l,.+l j. Letk hasr-adicdevelopmentk k.k_l

kl.

Then

()

0,(kY)’.

.’(kV)

o(k)

0,

aoe

.a, where

a

{0,1,

.,r-

1},s=

1,2,-

,)

(2.12)

From (2.10), (2.1 l)d

(2.12)

weget

(i)

0,

(aw,+l,1)

(aw+l,)j O,bb2. bj (2.13)

en0k<r

,

then0

(bib2.

.b)<r

dom(2.13)

wegetthatforljr-1

A

{(i):

jr

<

(j

+

1)r

"}

(m/r"+LO

m

<

r"+,m

j

(rood

r)}

The inequalities

(2.9)

areproved.

By

induction on thelenaisproved.

LE2.3 Let

((i)),o

bea

P-sequence.

Then for

eve

No

holds the equity

{(m

+

j)"m

O(mod r"),

0 j

<

r

}

{(m)

+

p(j)(mod

1)’m

O(mod r),

0

N

j

<

r

}

(2.14)

PROOF. Let us considerthat m kr

,

for some positive integer k. The equNity

(2.14)

is equiventtotheequity

{(m

+

j):m

O(mod r"),

0 j

<

r

}

r-1-1

t=o

{(m

+

j)’m

O(mod r"),lr

j

<

(l

+

1)r}

(2.15)

First,weshN1 prove that for

eve

fixedl,0 r-1ests uNquelyl"0 l"

<

r

"-,

such that {V(+j):=

,t

j

<

(t

+

1)

t

{(m)

+(j)(mod

1):m

kr",l’r

j

< (l"

+

1)r}.

(2.16)

Letk

(kk_l

k).

Thenwehave

v()

0,

r.

gn+ where for1

N

n

+

u

g

h=l

khVh+,,, (rood r).

Let0

g

<

r

-

be fix integer d

(/-1

l).

Then lr

(/-1

ll 0).

en

j is suchimeger thatlr j

< (l

+

1)r,

wehavej

(l_

lllo)r

where l_, ,1 e fixintegersd

l0

tes

rdifferent

vNues

inthet

{0,

1, r

1}.

t (j) 0,

a

.a, where

-1

(rood r)

a

l0

+

h=l

e

Vh+l, -1

(d r)

l

+

h=eh

Vh+l,e

av-1

1-2

A-

l-i

v,

-1

(rood r)

av

Iv-l.

Itisobviousthat andaltakesrdifferentvaluesintheset

{0,1,

r

1}.

where

(rn

+

j)

(0,

gTg

gg,,,+,

g,,,"+,,) *(

O,

a,a2

O, b]

b2

b,,,9,,+

b

gm +

a (rood r)

b-I

gv-1

+

av-1

(rnod r)

b2

g

+

a2

(rnod

r)

h

?

+

(.od )

(2.

7)

Since 0

_<

l"

<

r

"-1,

weshall search l’ in the form l"

(1"-1

/l")r,

wherell’,

.,

1",-1 are unknown quantities. Then l’r

(1’,_1

11" 0)r.

When l’ r

<

<

(l

+

1)r

then

(/’,-1

10"),

for 0

< 10 <

r.

Letusdenote

(i)

0,cl c2 c,where -1

(rnod r)

Cl

l0

-t-

’]h=l

lh

Vh+l,1

-

(rood r)

c2

ll

+

Eh=2

lh

Vh+l,2

Thenwehave

Cu-1

1"-2

Jc

l’u-1

Vu,u-

(rnod r)

Cu ,-1.

o(m)

+

o(i)(rnod

1)

O,

g

g".

gv-lrn g2

,q+lm

g,+nm

-

O, C C2" C,-1C

0, d

ld

du-ldu

gu+l "g+n where51, 62,

,

5u-1

arethestep-by-stepcarriesandelse

du

g’

+

cu (rood r)

du-

gu -Jr-

5u-

--

Cu-

(rood r)

d

g’ +52

+c2

(rood r)

dl

gr

+

51

._

(1

(rnod r)

(2.18)

For thedemonstrationof theequality

(2.16)

we makeequalthenumbers, constructedin

(2.17)

and

(2.1

$), andweget

l’u-1

=-

lu-1 (rood r)

l’u-2

+

l’u-1

Vv,u-1

+

5u-1

=--

lu-2

+

lu-1

Vu,u-1

(rood r)

u-1 -1

(rnod

r)

ll

+ ]’h=2

lh

73h+1,2

+

52

ll

--

Eh=2

lh

’Uh+l,2

-

-

(rood r).

10

--Eh=l

lh

’Oh+l,1

-

51

----l0

-Eh=l

lh

Vh+l,1

Since 0

<

lu-1, I’,,-1

<

r, then equation

lu-1

lu-1 (rood r)

has the only solution

1’-1

lu-1.

Consecutivelywe solvethe leftoverequations and getuniquely integer numberl’=

(l’u-

l’),

suchthat 0

<

I"

<

r

-1.

Since

l0

takesrdifferentvaluesintheset

{0,

1, r-

1},

then and

10"

takesrdifferent valuesin theset

{0,1,

r-

1}

and l’r

< < (l

+

1)r.

Finally,weestablishabijectionbetween thesetsfromthetwo sidesof the equation

(2.16).

Let.p

and q besuchthat 0

<

p,q

<

r

s,

p q and

p"

and

q’

arethenumbers, satisfying the equality

(2.16).

Weshallprovethat

p"

q’.

Letusadmit that

p"

q’

c. Thenwehave

{o(rn

+

j)"m=_0

(rnod r),

pr

<_

j

<

(p

+

1)r}

and

{o(m

+

j)"m=_₀

(rood

r),qr

<

j

<

(q

+

1)r}

{(m)

+

qo(i)

(rood 1)"m

=_₀

(rood

r),ar

<

< (a

+

1)r}.

Thenwehave

{o(m

+

j)’m=_₀

(rood r"),

_pr

<

_j

< (p

+

1)r}

{qo(m

+j)"m=0

(rood r"),qr <

j

<

(q

+

1)r}.

This is acontradiction,sincethefunctionpis aninjection;sothe equation(2.16)isproved. From(2.15)and(2.16)we get

rv-1

{o(m

+

j)"m=_0

(rood

r),O

<

j

<

r

}

J,.=o

-

{p(m)

+

qo(i)

(rood

1)"m

0

(rood

r"),

l’r

<

< (l"

+

1)r}

{(m)

+(j)

(rood

1)"m

0

(rood

r),O <

j

<

r"}.

The lemmaisproved.

3. AN ESTIMATION FROM ABOVE FOR

THE

DIAPHONY OF

Pr-SEQUENCES.

THEOREM3.1. Letintheguidingmatrix

(v,)

every

v,,,

1and for j

>

severy

v,,

0and let a

(o(i)),__

_obethe

P-sequence

which isproduced bythe

(v,).

Then foreverypositive integer Nwe have

Fg(cr)

<_

c(r)g-l(log((r

1)g

+

1))

1/2,

where theconstant

c(r)

isgiven by

c(r)

7r((r

1)/3

log

r)

1/.

(3.1)

The proof of this theoremisbasedon a non-trivial estimatefor the trigonometric sum ofanarbitrary

P,

sequence.

3.1. AN ESTIMATION OF

THE

TRIGONOMETRICSlIMOF

ARBITRARY

Pr-SEQUENCE. Let

X

,),,=0

is arbitrary sequence in interval E.A trigonometric sum,

SN(X:h),

of the sequence

X,

where his anintegeristhe quantity

SN(X;h)

-=0

N-1 exp

(27rihx).

LEMMA3.1. Let N

P

+

Q,

whereP and

Q

arearbitraryintegers. Then forevery integerh and arbitrarysequence

X

(x,),__

o wehave

SN

(X;h)

<

SP

(X;h)

/

Sp (X;h) l,

where

se

(x; h)

z_.,,=p exp

(27r/x,).

Theproofoflemmais obvious.

LEMMA3.2. Let N ar

,

wherea :>1andn

>

0are integers. Thenforeveryinteger hwehave

SN

(X; h)

<

E,=la

S(,_l)r,r-

(X; h)

Theproofof lemmaisbasedofLemma3.1and is donebyinduction ona.

Letabeanarbitraryinteger andq apositiveinteger. Wedefinethe function

q(a)

by

1,lfa=0(rnodq)

()

0.ifa0(mod q) Itiswell known that foreveryintegeraand every natural qwehave

q-1

E=o

exp(2riax/q)

q

(a)

LEMMA3.3. Let N :>1beaninteger and

g=’v__0ar

,ae{0,1,.

-,r-1}(j=0,1,.

-)

(3.2)

beits r-adic representation.

Letin theguidingmatrix

(v,)

every

v,

1and forj

>

s every

vs,

0andcr

(o(n)),__

₀be the

Pr-sequence

which isproductof

(v,).

Thenfor everyintegerhwehave

SN

(a;

h)

<

Ej:0

aj

r,

(h)

We shall prove that for every integer h and for every sequence X in interval E we have the estimation

[SN

(g; h)

<

:=0

Em=ll

Sire-l/r,

(S; h)[

(3.3)

where we have the supposition thatwhen

a

0, theinside sum is 0.

Let hbe aninteger. For

eve

N 1

ests

anintegern, such that N

<

r

.

Weshlprove the lena by theinduction onn

Ifn 1, then the estimation(3.3)istribal.

We suppose,that

(3.2)

istefor

eve

integer

N,

1 N

<

r

,

wherenissome integer.

Let now N such that r

N<r

+.

By

herewe have, that in(3.2)

a=0

forj>n Let

N

P

+

Q

where

P

ar

and

Q

a.

By

Lena3.1,Lena3.2and the induction suppositionweget

SN

(X;

h)

S:

(X;

h)

+

S

(X;

h)]

E=I

S:(m_):

(X; h)

’

a

E:0

E:

S(m_)

(x; h)

suchthat

(3.3)

isproved IfQ 0,then

(3 3)

isgotby Lena3.2.

Letnow j,0 j nbe

arbitra

fixed number and consider that 1 m

a.

Ifm 1, then by Lemma2.2for thetrigonometricsum

S

(a;

h)

wehave

s(;

h)=

,

(h)

(3.4)

(a;

h),

by Lena2.4, we have Letnow 2 m

a.

Thenfor thetrigonometricsum

S(m_)

(a;

h)

-x

k=0

-

exp

Sm_),

.:_),ezp

(2h(n))

(2ih(((m

))

d-1

+

()))

(ih((m

)))

E:0

zp

(ih(k)).

(a;

h)

we

et

Thus for themodule of the trigonometric sum

s_),

(; h)

,

(h).

(3.5)

From

(3.3), (3.4)

d

(3.5)

weget The lemma isproved. 3.2. PROOF OF THEOREM3.1.

SN

(a;

h)

S

Ej:0

aj

r

6r,

(h).

Let

(vs,)

is anarbitraryguiding matrix, such thatonprincipal diagonal there stand ones, and over him zerosanda

(,#,(n)),__

₀is

Pr-sequence,

which isbredbythematrix

(vs.).

WechooseN

>

1arbitrary integer and let hasr-adicrepresentationintheform

N

’],,

aa

rn,

(aj

_

{I,-

-,r-

I},

j 1,2,-

,k),

(3.6)

where

O

<

n

<

r2

<

<

nk.

areintegernumbers.

From Lemma

3.3forevery integerhwehave

Sw

(a; h)

<_

Ejk=1

ajr

,

5:,

(h) <

(r-

1)E3=1

r

,

5:,

(h).

By

thelastinequalityfor thediaphony

Fv(a)

ofawehave

(YFv(a))

2’h_-i

h-2

SN

a;

h)

g

:(

1)

E.:

-2(r

1)2

Eu:l

Eh=l

5:,

(h)6r-v (h).

If the matrix aj,

(1

j,

k)

is

setfic

then wehave

Byhered

(3.7)

weget

r

n+nv

h-2

(gf

())

a(-

)2

:

E:

E:

2(-

)

",

Eh:

h-2

:,(h).

Forj d such that 1 j k, wehave

(3.7)

(3.8)

forevery integerh. Beside this wehave

By

(3.$),

(3.9)

and

(3.10)

wehave

(NFu(a))

<_

(2 (r-

1)/3)

E:I

E:,

r"-"’-

(2/3)(

r

1)

2k

Forthe suminlast equityholds,that From (3.11)d

(3.12)

wehave

(NVu(a))

(/3)(r

-

X)

From (3.6)weget that Thuswediscover

k-1

N

>_

=r,

>_

’]=o

r

(r

k-

1)/(r-

1).

(3.11)

(3.12)

(3.13)

k

< (lo9((r- 1)N

+

1))/log

r

(3.14)

From

(3.13)

and

(3.14)

wehave

FN(a)

_<

7r((r

1)/3

log

r)

1/2N-1

(log((r-

1)N

+

1))

1/2.

The Theorem 3.1 isproved.

In the case, where theguidingmatrix

(v,)

is aunitmatrix

I,

thesequencewhich is bred by

I

is calledVanderCorput-Halton’s sequence. In 1935 it was firstintroduced by

Van

der

Corput [18]

and generalizedin 1960byHalton

19].

Inthiscase theoperation turns out tobeasimple addition.

By

o

(i)(i

0,1, wesignify thegeneraltermof theVanderCorput-Halton-sequence. Forr 2the sequence ofgeneralterms

o

₂

(i)(i

0,1, iscalledVander

Corput-scquenc.

By

Theorem3.1we canget thefollowingcorollaries.

COROLLARY 3.1. Leta

(o()),=0

be theVan derCorput-Halton-sequenc. Then for every positive integer

N,

wehave

FN(a)

<_

c(r)

N

-

(log((r- 1)N

+

1))

1/2,

where the constant

c(r)

isdeterminedbythe equality

(3.1).

COROLLARY3.2. Leta

((p(i))i__0

be theVander

Corput-sextuenc.

Then forevery N

>_

Iwe have

FN(a) <

aN

-

(log

N)

U2.

COROLLARY3.3. Leta

(o(i)),0

bearbitrary binary

P-sequence.

Then

ti--u,

(UFu(o))/ (o N)

in

<_

/

(o

2)in=

3,7773....

We notethat the Corollary3.1 and Corollary 3.2 are announced without proof byProinov and Grozdanov

[20]

andproved byProinovand Grozdanov

[21 ].

4. ON QUADRATIC

DISCREPANCY

OF

THE

SYMMETRIC

SEQUENCE

PRODUCED

BY

THE

ARBITRARY

P-SEQUENCE.

Inthissection,wegiven anappficationof Theorem 3.1 totheproblem of findinginfinitesequences in

E,

withthebestpossible order ofmagnitudeforthequadraticdiscrepancy.

We

need thenotionof symmetric sequence

(see

[11]). A

SeXluence a

(x,,),=0

in

E

is called symmetricofforevery integern

>

0wehavex2,,

+

x2,+ 1.

A

symmetricsequencea

(b,),=0

in

E

is said tobeproducedbyaninfinitesequencea

(a,,),__

0, iffor every integern

>_

0 weeitherhave

o

b2, or

a

b2,+1. Obviously, every infinite sequence in

E

produce at least one synunetric sequence.

By

Sobol

([5],

p. 117) is dear that the exact order ofquadratic discrepancy ofP2-sequeneeis

O(N

-

lo9 N).

By

thisandTheorem3 follows

THEOREM4 Let

’

be anarbitrary symmetric sequenceinE,whichisproduced byan arbitrary

Pr-sequence

Then foreveryintegerN

_>

2 we have

T.-(’’

<

c(r)N-t

(log(r

1)N))

/2

+

N

-1,

where

c(r)

is definedbytheequality(3 1) FromTheorem4 for thecaser 2 we have

lirn,w_,o

NT,

("

)/

(to9

N)/2

<_

1/(log

2)

1/2 1,201..., forevery symmetricsequence

’

produced bytheP2-sequence

We note that Faure [22] proved that for the symmetric sequence

’,

produced by the Van der

Corput-sequence,

theconstantlimN_,

(NTx(’)/(Io9

N)

1/2)

isbetween 0,298 and 0,321 REFERENCES

1. WEYL,

H,

Uberdiegleichverteilungvonzahlenmod.eins, Mat.Ann.77

(1916),

313-352 2. ROTH,

K.F.,

Onirregularities of distribution, Mathematika (1954),73-79.

3. ZINTERHOF, P., Uber einigeabschatzungenbeiderbei derapproximationyonfunktionen mitgleichverteilungsmethoden, Sitzungsber.Osterr.Akad.Wiss.Math-Naturwiss.Kl 185 (1976),121-132.

4. ZINTERHOF, P.

&

STEGBUCHNER,

H.,

Trigonometrischeapproximationmit gleichverteilungsmethoden, StudiaSciMath.Hungar 13

(1978),

273-289.

5

SOBOL,

M,

MultidimentionalQuadratureFormulas and

Haar

Function.Izdat "Nauka",

Moscow,

1969

6.

KUIPERS, L.,

Simpleproofofatheorem ofJK Koksma,NieuwTijdsch.Wisk. 55

(1967),

108-111

7.

NEIDERREITER,

H., Application of diophantine approximationtonumericalintegration, inDiophantine Approximationand itsApplication, C.F.Osgoot,Ed., 129-199. 8. PROINOV,

P D.,

Onirregularities

of

distribution,

C.R.

Acad.BulgarSci.39, No 9(1986),

31-34.

9. PROINOV,

P.D.,

Pointsofconstanttypeandupperbound for themeam-square discrepancy ofaclassofsequences, CR Acad.BulgarSci. 35

(1982),

753-755. 10. PROINOV,

P.D,

Estimation ofL discrepancyofaclass ofinfinitesequences,

C.R. BulgarSei. 36(1983),37-40.

11. PROINOV,

P.D.,

On

theL discrepancy ofsomeinfinitesequences,Serdica11 (1985),3-12. 12. SOBOL,

I.M.,

Onthe distribution of pointsin acubeand approximatecalculationof

integrals, J.Approx.Math &Math Phys. 7,No.4(1967),784-802.

13. SOBOL, I.M., Apoint/iniform distribdtioninmultididientionalcube,Izdat "Znani", Moscow1985

14.

FAURE,

H.,

Discrepances de suitesassociees a unsysteme de numeration

(en

dimention un),Bull. Soc.Math.France 109

(1981),

143-182.

15.

FAURE, H.,

Diserepancedesuites associees a unsysteme de numeration

(en

dimention

s),

Aeta

Arith.,XII (1982),337-351.

16.

NEIDERREITER, H.,

Pointsetsandsequenceswithsmall discrepancy, Mh. Math. 104

(1987),

273-337.

17.

NEIDERREITER, H,

Low-discrepancy and low-dispersionsequences, J.NumberTheory30 (1988),51-70.

$. VANDER CORPUT,JG, Verteilundsfunktionen,ProcKon.Ned. Akad. Wetensch.38 (1935),813-821.

19. HALTON,

J.H.,

Onthe efficiencyof certainquasi-randomsequencesofpointsinevaluating multi-dimensionalintegrals,Numer Math.2

(1960),

84-90.

20.

PROINOV,

P.D

&

GROZDANOV, V ST., Symmetrization of theVanderCorput-Halton sequence, C.R Acad.BulgarSci.40,No.8_(1987),5-8.

21.

PROINOV,

P.D.

&

GROZDANOV, V

ST.,

Onthe diaphony of theVanderCorput-Halton sequence, J.NumberTheory30

(1988),

94-104