On uniform distribution of sequences in GF[q,x] and GF\{q,x\}

FIq,

xl

AND

F

q,

1.

BY H. G. MEIJER

ANDA. DIJKSMA

1. Introduction and preliminaries.

Let

GF[q,

x]

denote the ring of polynomials in the indeterminate x over an arbitrary finite field GF(q) of q elements. If

A

and

M

are anytwo elementsof withdeg

M

_>

0,let

A (M)

be the uniquely determined element of such that deg

A (M)

_<

deg

M

and

A

-

A (M) (mod M).

J. H.

Hodges [2;

55]

definedthe uniform distribution ofasequence 0

(A)

ofelementsof asfollows.

Let

M

beanyelementof withdeg

M

m

>

0.

For

any

B

and integern

>_

1 define

0(n,

B,

M)

asthe number of terms among

A1, A2,

A,

such that

A(M)

B(M).

Then the sequence issaidto be uni]ormly distributed modulo

M

in if

(1.1)

lim

n-O(n,

B,

M)

q-"

for all

B

The sequence 0 is said to be uniyormly distributed in if

(1.1)

holds for every

M

withdegM m

_>

0.

Let

’

GF q,x denotethe extension field of consisting of all the.expressions

(c,

GF(q)).

Ifahas this representation and

c

0,thenwedefinedega m.

We

extend this definitionbywritingdeg0

.

The integral and fractionalparts of denoted by

[a]

and

((a))

respectively,aredefinedby

--1

E

iO

i--It

followsfrom thedefinition, that,for nd in

’,

wehve

[

-t-

f]

[-]

[].

We

say

(rood 1)

if

-t-

A,

where

A

.

It

follows thata

’

is

con-gruentmodulo1toaunique

,

namely

((a)),

such thatdeg

_<

0.

L.

Carlitz [1;

190]

defined the uniform distribution of a sequence

(a)

of elements of(’inthe following way.

For

any andanypositiveintegers

n

and/c,

define

O(n,/)

as thenumber ofterms amonga, a., a.suchthat deg

((a

fl))

<

-/c. Then the sequence is uni]ormly distributed modulo 1 in

’

if

(1.2)

limn-(n,

)

q-

forall k and

’.

ReceivedSeptember 16, 1968.

508 H. G. MEIJER AND A. DIJKSMA

(1.3)

C--. C--.--for infinitelymany s

_>

O.

An

elementa

’

issaid tobe irrationalif it isnotanelement ofGF(q,

x),

i.e.,

if itcannotbe written as aquotient

A/B

with

A

and

B

in

.

Well knownis

Kronecker’s criterion for irrationality" If x, then a isirrational if andonlyif

C-1 C-2

C_. _C-3 C_._

0

C_2+1

The aimof this paper istoextend someofthe results of

L.

Carlitzand

J. H.

Hodges.

To

do this we introduce a mapping of onto the set of nonnegative integers

I.

Let

r be a one-to-one correspondence between

GF(q)

and the set

{0,

1, q

1}

such that

r(0)

0.

We

extend the domain and range of to and

I

bydefining

r(anX"

an-X"--{- -]-alx2_

ao)

"r(an)q’*

-

T(an-)q

n-1

W

W

r(a)q

W

r(ao).

Clearlyris a one-to-one correspondence between

and

I.

Thenthesequence1

(C)

(r-(i

1))

consists of allelementsof

,

all occurring exactly once.

Hence

we have ordered the elements of

.

We

remarkthat F is uniformly distributed in (compare

[2;

62-63]).

S.

Uchiyama

[5]

has given a criterion for uniformdistribution ofa sequence in

I.

Usingthemappingr,we cangiveasimple criterion for uniform distribution in

.

See

₂

(Theorem 1).

In

₃

weprove that

(C)

isuniformly distributed modulo 1 in

’

ifandonly if a is irrational

(Theorems

2 and

3).

We

furthermore prove that

([C;])

is uniformly distributed in(I,_ifandonlyif isirrational or

A/B

with

A,

B

deg

A

_<

deg

B,

a 0

(Theorem 4). We

remarkthat

L.

Carlitz

[1; 191]

and

J. H.

Hodges

[2; 65]

have already proved that these sequences are weakly uni-formly distributed, i.e., they have proved that for these sequences the limits in

(1.1)

and

(1.2)

exist if n tends to infinity along the subsequence n

q(t

1, 2,

..-).

(We

notethatfor theconceptof"weakly uniformly distri-buted" defined in

[2]

is

not,

in general, the analog of this concept as defined for)’in

[1]).

Furthermore, weobserve that Theorem 4isthe complete analog oftIodges’Theorem 4.2.

2. Criterionforuniform distribution in

.

J.H.

Hodges

[2]

gaveanecessary condition for uniform distribution of a sequence in

.

L.

Kuipers

[3]

modified this condition to a necessary andsufficient one.

We

will givea somewhat less complicated criterion using the mappingr.

To

prove this criterionwe use the concept of uniform distribution in

I.

I.

Niven

[4]

defined this for a sequence

(an)

of elements of

I

asfollows.

Let

jandm

_>

2beanyelementsof

I

and define

I,(n,

j,

m)

tobethenumberof elements amongal, a2,

an

satisfying

a

-

(mod m).

Then the sequenceI,is saidtobe uniformly distributedmodulo

lim

n-lP(n,

,

m)

m-1 _{for all}

j,

I.

S.

Uchiyama

[5]

provedthe following criterion" P

(an)

isuniformly distributed modulomin

I

ifandonly if

(2.1)

limn-1 exp

(2wihai/m)

0 for h 1, 2,---,m-- 1.

For

the sakeof brevityweshallusethefollowing notation.

Let

M

be any

poly-nomial ofdegreem. Thenwedefine forany A,9andh,

_I,

eM(A, h)

exp

[2rihr(A (M))/q"].

TIEOIE

1. The sequence 0

(A) o]

elements

o]

9 isuni]ormly distributed modulo

M

in9_i]and onlyi]

limn-1

eM(A,

h)

0 for

=1

q--I

h= 1,2,

Proo].

Let

I,

_(r(A,(M)))

and

B

be an arbitrary element of 9. Then

A,

_=--

B (mod

M)is equivalent to

A,(M)

B(M)

or

r(A,(M))

r(B(M)).

Hence

P(n, r(B(M)),

_a

n)

0(n,

B,

M).

Thereforethe sequence0is uniformly distributed modulo

M

in9if andonlyif is uniformly distributed modulo

q

in

I.

Hence

Theorem 1 isadirectconsequence of

S.

Uchiyamds criterion

(2.1).

Thiscompletes theproof.

3. Uniform distributionof

(C,a)

and

([C,a]).

THEOIEM

2.

Let

r

(C)

(r

-(i

1))

andletcz 9’ be irrational. Then

the sequence

(Ccz)

isuni]ormly distributed modulo

I

in9’.

Proo].

Let

]c_{beanypositive integer and let}

___

_bx’

beanarbitrary

element of9’. Then since

._

cx

isirrational, there existsaninteger s

_>

k such that

(1.3)

holds. If

A

a,x

+

a,_x

-

+

+

no,withr

_>

s 1,

satisfiesthe inequality

(3.1)

deg

thenthe coefficientsa, a_,

ao

satisfy

aoc_

+

+

a._c_. b_

(a.c_._

+

+

ac_,_)

(3.2)

aoc_

+

+

a,_c_._+ b_

(a.c_._

+

+

ac__)

aoc-_

+

+

a._c_._ ei

(a.c_.__

+

+

ac___)

510 H. G. MEIJER AND A. DIJKSMA

where

e GF(q) (i

1,2,

...,

s

k)

arearbitrary. Ifs

],

thenthe equatiors of

(3.2)

containinge,vanish. Using

Cramer’s

rule, itfollowsthatwemay write

ao

Co.o Co.,a,

-+-

Co.ra,

a,_ c,-.o

+

c,_,.a,

+

+

c,_.a,.

where

_c.

GF(q). Here

the coefficients_C.o (i 0, 1, s

1)

dependon

ex

e., e,_, while the coefficients

c.

with j s, s 1, r are

inde-pendent ofel e, e,_.

Moreover

if

{e,

e,

e,_}

differs from

{e

e2

e,_}

thenthe corresponding set of coefficients

{Cg.o, C.o,

c.’_.o}

differs from

{Co.o,

Cx.o

c.-.o}.

Therefore the solutions

A

of

(3.1)

are of theform

A

_ao

X--I’

--1

(Co.o

-[-Cl.oZ -[- -}-c.-.o

-

a,(co.,

+

ci.,x

+

+

c,_.,x

-}-+

a,x

+

a,+,z

+

+

a,x.

Hence

(3.3)

A

where

F,, F,+I,

,

F,

arefixedpolynomials ofdegree

_<

s 1,a.,a,+1, a,

may be chosenarbitrarilyinGF(q) andwhere

G

isapolynomial with coefficients dependingone,e2,

...,

e,_. Since thereare

q’-

differentsets el, e2, e,_

thereare

q’-

differentpolynomials

G

and 1, 2,

q’-.

Now O(n, fl)

equals the number of polynomials umongC1,

C,

C

which areofthe form

(3.3);

i.e.,

O(n, fl)

isthe number ofpolynomialsofthe form

(3.3)

with

(3.4)

r(A)

r(a,)q

+

-

r(a,)q’

-}-

r(a,F

-

+

a,F,

₊

G,)

<_

n 1.

Suppose

firstthat

(3.5)

n-- 1 b,q’-l-b,_,q

-1+

+b,q’+

(q- 1)q

_"-1-...

-}-

(q-

1),

where 0

_

b <:

q- 1 (i s, s

-

1,

...,

r),

i.e., n

aq"

forsomeintegera. Since

{1,

2,

q’-},

we observe by comparing the equations

(3.4)

and

(3.5)

that

O,(n, fl)

q’-’(bq

-{- br_q

-’-1%

-t-

b.

-[-

1)

aq

Let

now nbearbitrary; then

fromwhichthe theoremfollows. This completes theproof.

THEOREM

3.

ais irrational.

(Ca)

isuni]ormly distributed modulo 1 in 0’ i] and only

Proo].

In

Theorem2 wehave shown that ifaisirrational, thentisuniformly distributedmodulo 1 in 0’.

Suppose

nowthata

A/B

where

A

and

B

belong

to andset deg

B

b.

We

may, anddo,suppose that

(A, B)

1. If t is

uni-formlydistributedmodulo 1in

’,

thenwe get from

(1.2)

withk b

-t-

1and

limn

-10,(n,

O)

Ifdeg((CA/B))

_<

-b 1,thereexistFandit (I,’such thatdeg

<

-b 1

and

or

CA/B

F

+

,

CA

FB

B.

Since

CA

FB

.

(anddeg

(B)

__<

--1,itfollowsthat/t 0and

B

divides

C.

Conversely, if

B

divides

C,

then deg

((CA

-

<

--b 1. Thus deg

((CA

<

--b 1 if and only if

C

0

(rood B).

Since the sequence

I’

(C)

isuniformlydistributedmodulo

B

in (compare

[2; 62-63]),

itfollows

that

lim

n-10b+l(n,

O)

lim

n-r(n,

O, B)

q-’

#

We

have thus arrived at a contradiction, and hence the theorem is proved.

TEOEM

4.

Let

r

(C)

beasabove. Then

([Ca])

isuni]ormly distri-butedin i] and onlyi]a is irrational ora

_A/B

where

A,

B 0, a 0 and a= deg

A

_

b deg

B.

Proo].

The proof is divided into three parts"

(I)

a is irrational;

(II)

a

A/B,

A,

B

,

anda

>

b;

(III)

a

A/B,

A,

B

,

anda

_

b.

I (a

is irrational).

Let

M

be any polynomial of degree m

_>

0. Then is irrationaland accordingto Theorem 2, O

(Ca/M)

isuniformly distributed

modulo1 in

’. Hence

ifD, withd deg

D

_<

m, thenfor/

_>

_0,

(3.6)

lim

n-’

e(n, D/M)

q-.

If

512 H. G. MEIJER AND A. DIJKSMA

then thereexist

F

and5

’

suchthatdeg5

_<

-kand

Ca/M

_DIM

F

"-I--or

Ca

FM

D

MS,

and hence

[C,o]

D (mod M)

if/c

>_

m. Conversely, if

[C,]

D (mod M),

then

(3.7)

holds fork m.

Because

ofthisequivalencewehavethat

O,(n, D/M)

(n,

D,

M).

From

thisand

(3.6)

itfollows that

lim

n-lI,(n, D,

M)

q-’.

II (

A/B;

a

>

b).

If

B

dividesC, then obviously

[C,A/B]

0

(mod A).

Converselyif

[C,A/B]

0

(mod A),

thenthere exist

F

and 5

’

such that deg5

<

0and

C,A/B

FA

or

C,-

FB

6B/A.

Sincedeg

6B/A

<

0,itfollowsthat

C

FB

or

C

0

(mod

B).

This implies that

[C,A/B]

0

(mod A)

if and only if C, 0

(rood B).

Since I"

(C,)

is uniformly distributed modulo

B

in

,

weget

lim

n-l,(n,

0,

A)

lira

n-IF(n,

O, B)

q-’

>

q-a,

whichimplies that the sequenceI,isnotuniformly distributed in

.

III (a

A/B;

a

<_

b).

By

definition

T(n,

D,

M)

isthenumber of elements among

C,

C2,

C.

which satisfy the equation

(3.8)

[XA/B]

=-

D(M)

(mod U).

Xo

satisfies

(3.8)

if andonlyifitsatisfies

(3.9)

[XA/B]

==-

D(M)

-1-

E,M

(mod AM)

q’.

We

nowdiscuss where

E,

is apolynomial of degree

_<

a, also 1,2,

foramoment equation

(3.9)

where and

D(M)

arefixed.

Let

_Xo

satisfy

(3.9).

Let F

be apolynomial ofdegree

<

b aandlet

H

bean arbitrarypolynomial.

Thenalso

(3.10)

Xo

+

HBM

+

F

isasolution of

(3.9). On

theotherhand,if

Xo

and

_X1

satisfy

(3.9),

then

[(Xo

XI)A/B]

=-

0

(mod AM).

Hence (Xo

X1)A/B

HAM

,

where ti 9 with deg5

<

0. Therefore

Xo

XI

HBM

B/A. We

set

F

B/A.

Thendeg

F

_<

b a, and since

F

_Xo

_X

HBM,

wehave

F

9. Thus if

(3.9)

hasasolution

Xo,

then

the other solutions are givenby

(3.10),

where

H

is arbitraryand

F

is arbitrary butdeg

F

<

b a.

Hence

there are

qb-a

solutions ofdegree b -{-m, andwe mayassumedeg

Xo

<

b

-

m.

Sincethereare

qb+m

polynomials of degree b m,itfollowsthat

q+: q-a

qa/"

equations of the form

(3.9)

are solvable.

On

the other hand, there are

qm

different polynomials

D(M)

and

q"

different polynomials

E,

so that there

are

q,n+a

different equations of the form

(3.9),

and hence allaresolvable.

Now

we want to determine

’(n,

D,

M),

the number of terms among

C1

C2,

Cn

which aresolutions of

(3.9)

for fixed and

D(M). In

otherwords,

wewanttodetermine the number ofpolynomials of the form

(3.10)

with

r(Xo % HBM

-

F)

<_

n 1.

Let

HBM

G

G.

where deg

G

<

b mand

with r deg

HBM

(if

H

0sothatr

,

then

_G

0).

Then

r(Xo

+

F

+

HBM)

r(dr)q

T

% r(d/)q

+ -{-

r(Xo

-

F

+

G).

Here

F

and

H

arearbitrary withdeg

F

<

b a.

BM

isfixed,while

_G

depends

onthechoice of

H. In fact,

ifwe compare the coefficients of

H,

BM

and

G,

weconclude thatthere isa one-to-one correspondencebetween the polynomials

H

and

_G.

Ifn eq

/’,

wegetasintheproofof Theorem2, that

,P(n,

D,

M)

nq

-’-.

Since

{1,

2,

q’},

weget

(n,

D,

M)

q"’(n,

D,

M)

nq-’.

From

thisitfollows aftersomecalculationthat

I’(n, D,

M)

nq-"

_

q

for all n, so that the sequence is uniformly distributed in 9. This completes theproof.

4. Complementarysequences.

Let

0

(A )

beasubsequenceof

r

(C)

(r-(i

1)).

If F, thenwedenoteby

_*

the complementary sequence of

,

which isasubsequenceof

r

and consists of allelementsof

r

whichdonotbelong to

.

Here

0* may be finiteorinfinite.

We

recall that

r

isuniformly distributed in9

(compare [2; 62-63]).

We

nowprove thefollowing theorem.

THEOREM

5.

Let

(A,)

be an

_infinite

subsequence

o]

F

(r-l(i

1)).

Let A(n, )

denote the number

o]

terms

_A

with

r(A)

<

n.

I]

s lim sup

n-A

(n, )

<

1 and isuni]ormlydistributedmodulo

M,

then

_*

isalso uni]ormly distributedmodulo

M.

514 It. G. MEIJER AND A. DIJKSMA

Proo].

Since lim sup

n-lA(n,

O)

<

1, the sequence 0*is infinite.

For

the

sakeofbrevitywewrite/1

A (n, 0)

and/2 n

A (n, )

A (n, *).

Then

kiln

<

(1

s)/2

and

k2/n

>

(1

s)/2

ifnis sufficientlylarge.

For

any

poly-nomial

B

wehave

O*(k,

B,

M)

r(n,

B,

M)

(/,

B,

M)

or

(4.1)

k;o*(].,

B,

M)

n-r(n,

B,

M)

+

(,/){n

-r(n,

B,

M)

7’(,, B, M)}.

Here

(//k.)

<

(1

s)/(1

s)

ifnis sufficientlylarge.

As

k

tendstoinfinity

through the sequence of all positive integers, then obviously n and/ tend to

infinity through subsequences of the sequence of all integers. Since

r

and 0 are uniformly distributed modulo

M,

the second term in the right-hand side of

(4.1)

tends tozero.

Hence

lim/c;

1*(k2 B,

M)

lim

n-’r(n,

B,

M)

q-’,

which provesthe theorem.

REFERENCES

1. L. CARLITZ, Diophantine approximation infields ofcharacteristic p, Trans. Amer. Math.

Soc.,vol.72(1952),pp. 187-208.

2. J. I-I. ttODGES, Uniformdistribution inGF[q, x], Acta Arithmetica, vol. XII (1966), pp. 55-75.

3. L. :UIPERS, A remarkonHodges’paperon_uniformdistributionin Gaoisfields (Abstract

presented by title), NoticesAmer.Math.Soc.,vol. 15(1968),p. 120.

4. I. NIVEN, Uniform distribution _of sequences of integers, Trans. Amer. Math. Sot., vol.

98(1961),pp. 52-61.

5. S.UCtIIYAMA, On the_uniformdistribution_ofsequencesofintegers, Proc. Japan Acad., vol.

37(1961),pp. 605-609.

TECHNISCHEI-IoGESCHOOL DELFT