SOME RELATIONS BETWEEN VARIOUS TYPES
OP NORMALITY OP NUMBERS

By
Henry Arthur Hanson

A THESIS
Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of
DOCTOR OP PHILOSOPHY

Department of Mathematics
1953

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SOME RELATIONS BETWEEN VARIOUS TYPES
OP NORMALITY OP NUMBERS

By
Henry Arthur Hanson

ABSTRACT
Submitted, to tne Scnool of Graduate Studies or Michigan
State College oi Agriculture ana Applied Science
in partial fulfillment oi the requirements
lor tne degree oi
DOCTOR OP PHILOSOPHY
Department or Mat nematics
±953
Approved by ______________ / 7 ^

- 1 The concepts of simple normality and normality of real
numbers, introduced by Borel in 1909, led to the intro­
duction of the analogous concepts of €-normality and (k,€)normality of integers by Besicovitch in 1934.

In this dis­

sertation certain relations are established between these
types of normality; the idea of simple normality is extended
to finite normality of order m; and the concept of quasinormality is introduced.
First, a number x, expressed in some scale B, is defined
to be finitely normal of order m in the scale B if every
sequence of m digits of the scale B occurs in x with the
asymptotic frequency of B~m .

It is shown by a constructive

proof that, for every positive integer m and for every scale
B, there exist numbers that are finitely normal of order m
but not finitely normal of any order greater than m.
Bext, the relations between e-normality of integers,
(k,e)-normality of integers, and normality of real numbers
are investigated.

It is shown that the problem of (k,e)-

normality of integers reduces to a problem of € -normality In
the following sense.

If an infinite sequence of increasing

positive integers is such that, for any given scale B and
any &>0, almost all of these integers are €-normal, then it
follows that, for any scale B, any positive integer k, and
any €>0, almost all of these integers are (k,€)-normal.

A

sufficient condition is established that such a sequence of

- 2 integers* when written in order and in juxtaposition after
the decimal point* shall form a normal number#

This suffic­

ient condition involves the rate of increase of the integers
of tne sequence.

It is shown by an example that if the in­

tegers increase too rapidly the resulting number may not he
normal#
Finally* a quasi-normal number is defined as a number y
which has the property that every number* derived from y by
selecting those digits whose indices form an arithmetic pro­
gression, is a simply normal number.

The question whether

numbers having this property are necessarily normal is ans­
wered in the negative*

This answer is obtained by the con­

sideration of a special class of quasi-normal numbers which
are constructed from normal numbers but which are themselves
normal only under a very special condition.

For this spec­

ial cl^ss of quasi-normal numbers a formula is developed for
the asymptotic frequency of any k-digit sequence.

1# Introduction
Emile Borel [2] introduced the concept of a normal num­
ber in 1909.

He first defined a simply normal number as a

number^ expressed in some scale B in which each digit of the
scale B occurs with an asymptotic frequency^ of l/B.

He

then defined a normal number as a number x, expressed in
o
some scale B* such that all of the numbers x, Bx* B x* • • •*
o
are simply normal in all of the scales B, B » B , • • •.
Borel further stated that the 'bharacteristic property"
of a normal number Is that every sequence of k digits*
k ■» 1* 2* 3,* * *, occurs in it with the asymptotic frequency of B

-k

•

This has quite generally been taken by subse-

1 The word "number" as used herein shall mean a decimal
fraction expressed in some scale

The word "decimal"

shall not mean that the scale is necessarily 10*
2

By the asymptotic frequency of a digit b in a number x

we mean
lim
n -too

Mn (x,b) ,
jj

where Hn (x*b) is the number of times b occurs in the first
n digits of x.

Similarly we mean by the asymptotic fre­

quency of a sequence* bjbg^^b^,
lim
_ —

k

in x*

Hn(x»blt>2,**'bk)
11

■

............. .

I I

•

quent writers as the definition of a normal number* However,
the proof of the equivalence of these two definitions wad
given only recently, in 1951, by I. Hiven and H* S. Zuckerman [9j •
In 1940, S* S. Pillai [lo] proved that a necessary and
sufficient condition that a number x be normal in the scale
p
B is that x be simply normal in all of the scales B, B , ••••
I)* D • Wall [llj , in his doctor’s dissertation, discussed
normal numbers from the viewpoint of their equivalence to
numbers x such that
one.

is uniformly distributed modulo

Wall proved, among other things, that if x is normal

in the scale B, then a x + b is normal in the scale B, where
a and b are rational numbers, a ^ 0.
Borel M

proved that almost all^ numbers are normal in

every scale B & 2 *

A short proof of this is given by Hardy

and Wright Q>,p* 126] • However, the first actual proof of the
normality of a particular number was published by D* G.
Champernowne £3j in 1933*
number

Champernowne proved that the

.1234567891011 • • • is normal in the scale 10. It

When applied to the numbers in an interval, almost all
means all except a set of measure zero. When applied to a
denumerable sequence .£an '^, it has the following meaning:
Almost all an have the property P if the number among the
first n elements which fail to have the property P is o(n)
as n-^00*

- 3 is easily proved, "by the Copeland-Erdos theorem referred to
below, that a similarly formed number^in any scale B is
normal in the scale B*
introduced the concept of
(k,€*)-normality, which is an extension of the idea of norm­
ality to positive integers.

where the

0,^

An integer

are digits of some scale B, is (k,€)-normal in

the scale B if, for every k-digit sequence, ^ 1^ 2*

where M(m,b;|br,#•’b^) is the number of occurrences of
b-jbg*,#bk

in m.

6-normal*

Besicovitch proved that, given any integer k ^ l ,

any

If k ~ 1* we shall say simply that m is

€>0, and any scale B, almost all positive integers are

(k, 6)-normal in the scale B.

He also proved that, given any

6 > 0 and any scale B, almost all squares of integers are
6-normal in tne scale B*

4 This number is formed by writing the set of all positive
integers consecutively in juxtaposition after the decimal
point*

Generally, we mean by the number .a^a^a^* • •, where

{an> is a sequence of positive integers expressed in some
scale B, tne decimal formed by writing after the decimal
point, in order and in juxtaposition, the integers an .

- 4 Using Besicovitch1s concept of (k,c)-normality* A. H.
Copeland and P. Erdos [4] proved, in 1946, that if £an} is
an increasing sequence of integers, then .aia2a3 » • • is a
normal number, provided the number of a.±'s less than S' is
greater than ITe for every 0 < 1 and all K>lTu (e),
H. Davenport and P. Erdos
of the following two theorems.

published in 1952 a proof
(i) If f(x) is a polynomial

all of whose values for positive integral values of x are
positive integers, then the number
normal.

.f(l)f(2)f(3) • • *

is

(ii) Por any £ and k, almost all of the integers

f(l), f(2), f(3), • • • are (k,€)-normal.

Actually, with

little more than a change of notation, their proof holds for
every f(x) whose leading coefficient is rational, so that,
with only this restriction, the results stated hold with
instead of f(n).
We shall now describe briefly the results obtained In
this dissertation.
In Section 2 we shall introduce a type of normality of
numbers which is an extension of simple normality. A number
x, expressed in some scale B, we define to be finitely normal
of order m in the scale B if every sequence of m digits of
tne scale B occurs in x with tne asymptotic frequency of B~m .
It Is obvious that, if a number is finitely normal of order
m, then it is finitely normal of every order j < m .

We shall

prove that, for every positive integer m and every scale B,

- 5 there exist numbers that are finitely normal of order m in
the scale B, hut not finitely normal of any order greater
than m.

This we shall do by demonstrating a method for the

construction of sucn numbers*
In Section 3 we shall he concerned with some relations
which exist between ^-normality of integers,
of integers, and normality of numbers*

(k,£)-normality

We shall show that

the problem of the (k,€)-normality of integers reduces to a
problem of e-normality in the following sense.

If an infin­

ite sequence of increasing positive integers is such that,
for any given scale B and any €>0, almost all of these In­
tegers are e-normal, it follows that, for any given scale B,
any positive integer k, and any

€>0, s,lmost all of these

integers are (k,€)-normal*
[Following this, we snail prove a sufficient condition
that such a sequence of integers, when placed in order and
in juxtaposition after tne decimal point, shall form a norm­
al number in the scale in which the integers are expressed.
This sufficient condition involves the rate of increase of
the integers of the sequence.

An example will be given to

snow that, if the rate of increase is too great, the result­
ing number may not be normal*
In Section 4 we shall be concerned with quasi-normal
numbers.

We define a quasi-normal number as a number y

whicn has the property that every number derived from it by
selecting tnose digits whose indices in y form an arithmetic

- 6 progression is a simply normal number.

The question whether

numbers having this property are necessarily normal will be
answered in the negative.

This will be done by means oi a

special class of quasi-normal numbers constructed from norm­
al numbers*

The quasi-normal numoers thus constructed, we

shall prove, are normal only under a very special condition.
i?‘or these quasi-normal numbers derived from normal numbers,
a formula will be given for the asymptotic frequency of any
k-digit sequence.

- 7 2. Finitely normal lumbers
We defined a finitely normal number of order m in
Section 1.

It is obvious that a number which is normal in a

scale B is finitely normal of oraer m in the scale B for ev­
ery positive integer m.

We are interested here in une con­

struction of a number which, for a given scale B and a given
m, is finitely normal oi oraer ra, but not finitely normal of
any order greater than m.
LKM m A 2.1.

Given any scale B and any positive integer

there exists an integer Im having j>m + m - 1 digits in cue
scale B, such that every sequence of m uigits of the scale B
5
occurs in Im exactly once.
PROOF. An integer Im satisfying the lemma is constructed
as follows. Write m successive digits (B-l), and follow
these at each successive position by the smallest digit in
the scale B which does not cause the repetition of a pre­
viously occurring sequence of m digits. Continue the con­
struction thus until no longer possible.
We shall show that the integer Im thus constructed has
3 m + m - 1 digits and contains every sequence of m digits
exactly once.

The method of construction assures us that no

m-digit sequence is repeated in Im and, therefore, it is

& This lemma is an extension of a problem solved by Roger
Lessard, American Mathematical Monthly, vol.58(1951),p.573.

- 8 sufficient to prove that every m-digit sequence occurs.
The integer Im ends in m-1 successive digits (B-l). For,
suppose that the last m-1 digits contain some digit other
than (B-l). Then this sequence of m-1 digits must have oc­
curred previously followed by each of the digits of the
scale B; and, including this last occurrence, it must have
occurred B+l times.

Since these m-1 digits are not identic­

al with the first m-1 digits of I , each of these occurrencm
es must have had a distinct preceding digit. But this Is im­
possible. Hence the last m-1 digits of Im are all (B-l)Ts.
The sequence of m-1 digits (B-l) occurs in Im exactly
B+l times, the first time without a predecessor, the other B
times preceded by each of the digits of the scale B.Gonsider
now an arbitrary fixed m-digit sequence,b^B-l)(B-l)•••(B-l),
where b^ ^ B-l.

Before the occurrence of this m-digit se­

quence, the m-1 digit sequence, b^(Bi-l) (B-l) *•• (B-l), must
have occurred B-l times, and nence this latter sequence must
be preceded in Im by each of the digits of the scale B. So
every m-digit sequence of the type, bgb^B-l) (B-l) • •*(B-l),
where b^ and

\>2

are any digits of the scale B, b^

B-l,

must occur in Im .
By an obvious induction we see that every possible
m-digit sequence occurs in Im ; and since each occurs exactly
once, the number of digits in Im is 3 m + m - 1. This com­
pletes the proof of the lemma.

- 9
THEOREM 2.1. Given any scale B and any positive Integer
m, there exists a number which is finitely normal of order m
in the scale B, but not finitely normal of any order greater
than m.
PROOF.

Let the first B111 digits of the integer Im of

Lemma 2.1. be taken as the period of a repeating decimal. The
resulting number is clearly one in which every sequence of m
digits has the asymptotic frequency B~m .

It is evident also

that the number thus constructed is not finitely normal of
any order greater than m, since there are certain sequences
of m+1 digit's which do not occur in it at all, for instance,
the sequences of m+1 equal digits.

This completes the proof

of the theorem.
The finitely normal number of order m constructed in the
proof of Theorem 2.1 is a rational number.

This number,

however, can be modified in such a way as to become irration­
al without disturbing its finite normality.

For example, we

could arbitrarily change every digit whose index is a power
of 2.
We note that, for finitely normal numbers of order m
which are not finitely normal of any order greater than m,
the fact that every sequence of m digits occurs with the
asymptotic frequency of B"m does not imply that the m-digit
sequences occur with equal asymptotic frequency in the m

- 10 possible positions

modulo m* The finitely normal number of

order m constructed in the proof of Theorem 2*1 is such that
every m-digit sequence occurs with equal asymptotic frequency
in the m possible positions modulo m if and only if B and m
are relatively prime*
This remark is of particular interest in connection with
the result of Kiven and Zuckerman

mentioned in Section 1*

Their result amounts, essentially, to showing that if every
m-digit sequence occurs in a number with the asymptotic fre­
quency of B “m , and if this is true for all m, then the oc­
currences of each m-digit sequence are also equally distrib­
uted asymptotically among the m possible positions modulo m*
The above remark points out that this conclusion cannot nec­
essarily be drawn if the statement that every m-digit se­
quence has the asymptotic frequency of B~m is true only for
m not greater than some mQ«
For example, if B ■ 2 and m — 2, the finitely normal
number of Theorem 2*1 is
.11001100 • • •,
in which each sequence of two digits occurs always in the
same position modulo 2*

^ We will say that a k-digit sequence occurs in a position
congruent to i modulo j if the index of the first digit of
the sequence is congruent to i modulo j.

11 3* Some Relations "between €-Normality,
(k,€)-Normality, and Normality
When Besicovitch proved that the squares of almost all
integers are £-normal, he did not extend the proof to (k,€)normality*

The particular argument which he employed to pro­

ceed from ^-normality to (k,€)-normality in the case of the
set of positive integers breaks down when applied to the
squares oi' integers*

The fact that the squares of almost all

integers are (k, €)-normal is, of course, only a special case
of the second Davenport-Erdos theorem [5,p.58].

We shall

prove, in what follows, how, in a quite general way, the
problem of (k, €)-normality reduces to one of e-normality*
LEMMA 3*1*
ient condition

(D

Given €>0, and an integer k ^ 2 *

A suffic­

that an integer

m = a^-iVs

* * * a la0

be (k, €)-normal in the scale B JUs that m be £T-normal in the
scale B r , where k/r<€/3, €*Br< 6/3, and that m be sufficiently large so tnat r/jlk.< €/6B^m
PROOF*

Let m be an integer which is €* -normal in the

scale B r , wnere r, £*, and

satisfy tne inequalities of the

hypothesis*
The digits of the scale B , when represented in tne scale
B,constitute the complete set of r-digit sequences

of the

scale B, (if we write, where necessary, initial zeros)* Let

- 12 *>k be any given sequence of k digits in the scale 3.
The sequence l3ib2 ***bj£; occurs in the complete set of r-digit
sequences of the scale 3 exactly ( r -k+l)Br~^ times*
Let us represent m in the scale B r ,
(2)

m as

where ju/r £

V

• • AjAq,

=f: 0),

< ^c/r + 1.

Let us then replace each A i hy the corresponding sequence
of r digits in the scale B*

We obtain in this way a repre­

sentation of m in the form,
(3)

m =■ UQ * • * OgL^.ja^.^ • • • a^ag,

where the number of initial zeros (of which there may be
none) is less than r*
Since m is € f-normal in the scale B r , every digit of the
scale B r occurs in (2 } more than (B“r- € * )V" times*

Hence

bibg***b^. occurs in (3) more than
(B"r- e')*!±.(r - k + l)Br-k
times. In this estimate we disregard possible occurrences of
lolk>2 **#fc)k beginning in some Aj^ and ending in

we

also take account of the fact that, in going from one repre­
sentation of m in (3) to the representation of m in (i),
less than r sequences of k digits can be lost, we see that

- 13 ^ (B“k - e ’B r'lc)/K ( l >(iT* Bk

rBk

- r

. r)
>*■ ~

>(B"k - -%)(>*•- k + 1).
Bk '
Tills is true ior every k-ctigi& sequence. It follows,
then, that for every k-digit sequence, b-jbg* ••h.^,
ISrCm.Tj^g* .*l3t ) <

(B-* -t- € )(yM. - ]£ + 1) ,

and hence m is (k,€)-normal in the scale B.
THEOREM 3.1. Let

be an infinite sequence of in­

creasing positive integers having the property that, for any
given O

0 and any given scale B, almost all of the integers

a^, a«, a^, * • •

are €.-normal.

the property that, for any given

Then the sequence {* n> has
€ > 0 , any given k £ 2 , and

any given scale B, almost all of the integers a^,ag,ag,» • •
are (k.€)-normal.
PROOF.

For a given scale B, a given

€>0, and a given

k, choose an r and an € ’ which satisfy the hypothesis of
Lemma 3.1.

Then, for a given

*|>0, choose an

such that

for all n > B ^ , more than (1 - >]/2 )n of the integers a-^, ag,
• •

agi are 6*-normal in the scale B r .

Choose also an

Bg such that, for all n > B g , wore than (1 - f)/2)n of the
integers a^, ag, * * •,
B =

max (B^, Bg).

are such that ^A>3rB^/fe .

Let

Then, for all n > B , more than (1 - Kpn

of the integers aj, ag, ag, • • •, are (k,€)-normal in the

- 14 scale B#

This completes the proof of the theorem#

How let us consider a sequence £an^

increasing pos­

itive integers expressed in some fixed scale B, such that,
for any given k and any given

£ > 0 , almost all of the integ­

ers a^, ag» aj, • • • f are (k,€)-normal in the scale B# We
inquire about the normality of the number •
the scale B#

• •

in

The following theorem furnishes a sufficient

condition, based upon the rate at which the integers a^ in­
crease, for the normality of the number
THEORBM 3# 2#

Let

• • • •

be an increasing sequence of pos­

itive integers having the property that, for any given k and
any given

€> 0 , almost all of the integers a^, ag 9 £3 ,• • *,

are (k, Q - normal

inthe

of digits in a^,

i

1,

scale B#

Let

denote the number

2, 3, • • *, and let

Sn = < g L V i *
Tnen a sufficient condition that the number x = •a1asa3 » • *
be normal in the scale B in that
(4)

nVn = 0 ( S n ).
PROOF#

Let

of the scale B.
Sn < m < Sn+1#

be any given sequence of k digits
Let m be an integer, and let n be such that
Then for a given
>

£>0,

(B-k - O f ' K

- * + !).

where

is taken over the values of A4 n for which a^ is

(k, C) -normal.
Let the number of the integers among a^, a^, • • •> an >
which are not (k,£)-normal be denoted by d>n .
ft)n s o(n) as n-*oo.

By hypothesis

Also V ^ Y n for every \ 4 n.
^
>

(3 “k " e)-fe v *

Hence

- (n - wn )(k-l)J

(B"k - €)(Sn - <UnVn - nk),

and
HmCxjbjbg* •*b^)

------—

m - k + 1

—

Sn - ^nvn ” nk

> (B

- C)

-------

sn+l

. (B"k - e)fl - -Jr . (n-H)Vn+l\/j . «n.nVn - k.,ny^\
L
S n+1
j \
n Sn
7 Z Sn }
t \
wnicn, by (4), approaches B

- € as n-*oo.

Hence
lim
m-*«o

B m U t B i ^ ’^ k )
m -k + 1

>

B"k - € ,

and, since e can be taken arbitrarily small,
lim
m-foo

^ ( x ^ y ^ f c )
m — k *f" 1

v „-lc

Since this is true for every kfdigit sequence, we have
lim ”k(*»blV»««>k)
m-foo
m -k + 1

_k
*

and x = •a1a 2a3 * • • is normal in the scale B.

- 16 REMARK 1.

A sufficient condition

Yn 5=1 0(3-°g n).

tor (4 )is that

For, noting that

Yn = 1 t [logB a j >

° »

we see that
nYn ^
Sn

(G log JS)n log n

(C log

B)n log n

log 2 4- log 3 4- •• • + log n ~ n log n 4- o(n log nT *

wnicn approaches C log

3

as n->oo.

Tne condition, V n = 0(log n), is certainly satisfied if
jjr(n)] , where f(x) is a polynomial.

a,, -

degree ot f(x).

For, let h be the

Then an ** Ofn*1), and Vn as 0(log n).

This fact can he used to simplify tne proof of the two
theorems in the Davenport-Erdo's paper £o,p.08j . The two the­
orems are there proved independently.

By our result,

the

lirst theorem of tne Davenport-Erdos paper is a consequence
01 tne second.
REMARK 2.

Another case in which (4 ) no Ids is that in

which tne integers

increase exponentially5 more generally,

a suAiicient condition for (4) is that
^ n * < Yn <
where

Sp

JW-2* 811(1

*

n = 1, 2, 3, • • •,

are positive constants.

•+

Ml

Here we hare

n01^ / ^ ! ) + o(nW + 1 )

- 17 so that n Vn/sn is "bounded#
KIMARK

If> however, the an increase too rapidly, the

conclusion of Theorem 3*2 does not necessarily hold# Vie con­
struct now an example of this kind#

Consider the number of

occurrences of any given sequence, ^ 1^ 2 ***^*
the scale B, k < m ,

^ digits of

in the integer Im of Lemma 2#1#

^1^2*' ,l5k *

If

then Ndmfbjbg* •*bk ) = B,:a"k .

If h**2* *

“ (B-l) (B-l) • •• (B-l), then N d m . b j b ^ •-bk ) =

B®1' + m - k.

Kence we have, in any case,
irdm.bibg*"!^) = Bm-k +

e(m-k),

where 0 = 0 or 1#
It follows that
H d a ,bib2 »»«bk ) _ 1
T>m
B
15 - k + m

(5)

|e(m-k)B^ - (m-k)l
(Bm *f m - k)B^

v
^

Bm + m - k
(m-k)(B - 1)
(Bm + m - k)Bk

Hence, for any given k and any given

£^0, almost all

Im , (in fact, all except finitely many), are (k,£)-normal*
How let Jm = lm if m is not the square of an integer, and
let Jm = (B-l) (B-l) • •♦ (B-l), where the number of digits is
B m+ m - 1 ,

if m is the square of an integer#

quence

Then the se­

k as t*16 property that, for any k ^ l and any

£>0,

almost all of its members are (k,S)-normal. (Note that, for
the sequence

rate of increase of Vn , defined as in

Theorem 3*2, is like that of a geometric progression.)
prove that the number x =* •

x. d o

Vfe

• • fails to be normal•

To prove this it is sufficient to show that x is not simply
normal*
In the notation of Theorem 3*2, i f m i s

the square of an

integer,
Ng^x.B-l) > Bm + m - 1 £ Bm ,
and
K Sm(x,B-l) ^
Sm

B+(B2+ i )+(b 3+2)+«--+( b " W - 1 )
m
3
(Bnrt-1_B )/(B _ i

)

^ Bm (B-l)
; B-l .
imtl Bm^
B

1

,

so that
lim
n^oo

-

*

B-l
—
*

This proves that x is not simply normal if B > 2 .

In

the case of B = 2, we need a better estimate of Bg^Cx,!).
Making use, then, of the fact that more than half of the
digits in each 1^ are I ’s, we have, if m is a square greater
than 1,

- 19
This estimate leads to
lim Hn U , l ) ^ 3 .
n-»oo
n
^ %
Mote that relation (5) of the above example furnishes us
also a method for the construction of an integer in a given
scale B which, for any given k and any given £>0, is (k,G)normal in the scale B*
nv/B m <€

We need only choose an m such that

and m ^k, and then the integer Im is (k,6 /-normal

in the scale B*

-

20 -

4* Q,uasi-Hormal Numbers
In Sectional we defined a quasi-normal number as a number
y, expressed in some scale B, having the property that every
number formed by selecting those digits of y whose indices
form an aritnmetic progression is simply normal In the
scale B.
Every number which is normal in the scale B is also
quasi-normal in the scale B.
below.

This follows from Lemma 4*1

This lemma is a generalization of a direct conse­

quence of Borel’s definition of normality (see p.l), namely*
that if x is normal in the scale B, then every sequence of k
digits of the scale B occurs with equal asymptotic frequency
in each of the k possible positions modulo k. (See footnote
6, p.iu).
LEI/IMA 4.1.

If x ijs a normal number in the scale B, and

k, j , and i are any positive integers, and ^x^2**#^k —

any

sequence of k digits og the scale B; then l>ib2***^k occurs
in x in a position congruent to i modulo j with the asymp­
totic frequency ox i/jB^.
PROOF.

Bet r be the smallest integer for which r j ^ k .

Every sequence of rj digits occurs in x in a position con­
gruent to 1 modulo rj with the asymptotic frequency ox
r i—k
l/rjBr<*. Among these sequences of rj digits each, B
be­
gin with t)ib2**#1:)k*
occurrences in x of

Hence tne asymptotic frequency of the
I*1 a position congruent to 1

-

modulo rj is l/rjB •
the numbexSiB

P-I

-

Applying the same principal to any of

x, wnere

sult that, for a fixed

21

i,

2,

•*•, rj, we obtain the re­

l £ p 4 rJ* the asymptotic frequency

ox trie occurrences 01

in x in a position congruent

to p modulo rj is also l/rjB^,

Since m e r e are r values of

p that are congruent to i modulo j , it follows that the
asymptotic irequency of the occurrences of

in a

position congruent to i modulo j is l/jBk *
The statement that every normal number is also quasinormal is merely tne special case of Lemma 4*1 for k a 1#
We are led to inquire whether the converse of this
statement is true - is every quasi-normal number also normal?
This is not true, as we shall prove in the investigation of
a special class of quasi-normal numbers*
THEOREM. 4*1*
scale B*

Let x be a number which is normal in the
7
Let s be any integer greater thgn 1* Let

rj = res [B J*] (mod s).

Let n . denote the number of digits

preceding the j^*1 occurrence of any given k-digit sequence,
bfbg^^^b^ in x*

Then

? If X is any real number and q is a positive integer,
then we mean by
0 £ <r < q

res X (mod q ) , the number

and (X - <r)/q is an integer.

cr, where

-

22 -

(*■) the number y = .r^rgrj* • •

ij3 quasi-normal in the

scale s;
(ii) the number *rnirn2rn3* * *

simply normal in the

scale s.
PROOF.

By Wall’s result

[ll] , x/g is normal in the scale

B, and ^Bnx/s^ is uniformly distributed modulo 1.
Tne number

res B nx/s (mod l), which is the fractional

part of the number B n;x/s, falls into one of the s intervals*
(Q,l/s), (l/s,2/s),* • • , (1-1/s,1), namely, into the in­
terval (°/s, (cr+ l)/s), where O'= res£BnxRmod. s).
{b V

s}

Since

is uniformly distributed modulo 1, the number of

numbers res B nx/s (mod l) in each of the above intervals is
asymptotically equal to iy^s, and, consequently, the integers
res[jBnxJ(mod s) are asymptotically equally distributed among
the integers 0, 1, 2, • • • , s-1.
y = .r^rg^.

Hence the number

• .is simply normal in the scale s.

Further,

the fact that x is normal in the scale B imu—t
t
plies that B
x is normal in the scale B , where u and t
are any positive

integers.

Let us take u ^ t .

x zz .a^a^a^* • •

in the scale B,

Then if

B u "^x = .000*••Oa^aga^* * *
in the scale B, where the number of initial zeros is t-u.
_
t
In the scale B ,

Bu-tx = .AjApAg • • •,
where A 19 represented in the scale B, is 000* •‘a-j^a^* •*au ;

- 23 ana A., j >1, represented in the scale R, is
J

au-*-(j-2)f+lau+(j-2)f*2* * * au+(j-l)t#
Ix we write
Rj * res(Bt*t;(jBu "t:jj! (mod s),
then, by tne preceding argument,
al in tne scale s#

• • Is simply norm­

Since Rj as

j-l)t»

"the number

#rur u+tru*f2tru+5t* * * Is simply normal in one scale s,
y =* •r ir2r3 # • •

and

is a quasi-normal number in the scale s#

For the proof of (ii), consider those values of n for
which the numbers res B nx (mod 1) lie in the interval of
length I/b ^" whose left end point is

^°'te that

these are precisely the values n^> Ug, n3 ,• • • , defined in
the statement of the theorem*

For each of these values of n,

the number res B nx/s (mod 1} lies in one of s non-overlapping
intervals of length l/sB^#

Of these intervals, one lies in

each of the intervals (^/s, (s'+lj/s); and, for each n, the
value of O' is determined by O'= res[Bnx](mod s).

Since

there are asymptotically n/sB^ numbers res 3 nx/s (mod l) in
each of these intervals of length l/sBk , it follows that the
values of resjBnx](mod s) for the values n x, n g, n3 ,» • •»
are asymptotically equally distributed among the integers 0,
1, 2» • * #t s-1; and, therefore, the number •rn x:rn2 • • •
is simply normal in the scale s.
of the theorem#

This completes the proof

- 24
REMARK.

Hote that it follows from Theorem 4.1 that the

asymptotic frequency of a given digit of tie scale s in y in
the positions n ^ » as defined in the theorem, is l/sBk .
The number y, derived from a number x which is normal in
the scale B, in the manner described in Theorem 4.1, is
quasi-normal in the scale s.

The additional property (ii)

which the number y possesses enables us to calculate the
asymptotic frequency of any given k-digit sequence in y, and
thus to determine whether y is normal in the scale s«
THEOREM 4*2.

Let x be normal in the scale B; let s be

an integer greater than 1; and let y
k-digit sequence in the scale s.
quency of Xlf2* *

^

y

5®. £ given

Then the asymptotic fre­

iH the number y, as defined in

Theorem 4.1, ijs equal to

Here

S±

=

0 or 1, according as

or yUjL<m, where

m = res B (mod s) and yAj_ = res (^i - ^..iB) (mod s)«
PROOE.

Let c^Cg***©^ denote any sequence of k digits of

the scale B, and let r denote the integer res[jBnxJ(mod s),
where n is the number of digits in x preceding an occurrence
Of

C iC g***^.

We inquire, first, how many combinations consisting of a

-

25

-

value of r and a sequence c ^ C g * * * ^ in x w m

result in the

occurrence of the given sequence $1)2 * *#Yk in tile correspond­
ing position in y*
For each of the digits of the sequence c^Cg**^^. and each
of the values of r, the following relations must holds
(6)

rB + c-j, = K i (mod s)

8lB + c2 ^ ^ 2 ( mod
+ c3 =2fe(mo<i B )

(7 )

tfk-lB 4- Ck = y k (moa s)
where 0 ^ r < s >

0 ^ c i <B.

The left member of relation (6) takes on sB values, 0, 1,
2f * • *, sB-1, as r and c^ range independently over the in­
tegers from 0 to s-1 and from 0 to B-l respectively. Of these
sB values of the left member of (6), exactly B values are
congruent to

modulo s.

The relations (7) are independent of each other and of
(6).

For the relation, Yi»is t1 c ^ = ^ ( m o d s), it is easier

seen that there are either [b / s]+ 1

or Q^/s] values of c^

which satisfy this congruence, according as m>yix1 or
where m a n d a r e

defined as in the statement of the

theorem#
Since the relations (6) and (7) are independent of each
other, the number of combinations of values of r and se-

- 26 quences ciC2 ***ck which result in a given sequence
#*Yk in y is

where 5^ is defined as in the statement of the theorem.
From the remark following Theorem 4.1* it follows, then,
that the asymptotic frequency of

in y ls equal to

REMARK. If we define an empty product to he 1, then the
formula of Theorem 4.2 gives also the asymptotic frequency
of each digit of the scale s in y as l/s.
THEOREM 4.3. The number y fs normal in the scale s if and
only if s divides R*
PROOF. If s divides B, then the value of m of Theorem 4.2
is zero and each factor of the product j~J is [b/s[ , which is
equal to R/s.
quence

Thus the asymptotic frequency of any given se­
in y is

If s does not divide B, then consider the asymptotic
frequency of any k-digit sequence, yife* *'tfk9 in

- 27 -

In order tnat this "be equal to l/s^, we must have

But the left member of this equation is an integer,
While the right member is not, unless Is: s 1. Hence if s does
not divide B, y Is not normal in the scale s.
Thus Theorem 4*3 answers in the negative the question
whether a quasi-normal number is necessarily normal* Indeed,
with regard to the class of quasi-normal numbers y, derived
in this manner from a normal number x> we can say that if
s aoes not divide B, tnen icr no k > l does any sequence of
k digits have the proper asymptotic frequency, I/sfc. We no be,
too, that if s > B ,

then by (7) there are some sequences of

digits of the scale s which do not occur in y at all, for
example, any sequence, $ 1 ^ 2 * in which a zero is followed
by a digit equal to or greater than B*

-

28 -

BIBLIOGRAPHY
1# A* S. Besicovitch, The asymptotic distribution of the
numerals in the squares of the natural numbers,Mathematische
Zeitschrift, vol. 39 (1934), pp. 146 - 156.
2. E. Borel, Les probabilites denombrable et leur applic­
ations arithmetiques, Rendiconti del Circolo Matematico . di
PaLermo, vol. 27 (1909), pp. 247 - 271.
3. D. G. Champernowne, The construction of decimals norm­
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8. J. E. Maxfield, A short proof of Pillaits Theorem on
normal numbers, Paciiic Journal of Mathematics, vol. 2
(1952), PP* 23 - 24.

-

29

-

9. I. Miven and H. S. Zuckerman, On tne definition of
normal numbers, Pacific Journal of Mathematics* vol. 1
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••

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