Irrationality measure of sequences

Abstract

The new concept of an irrationality measure of sequences is introduced in this paper by means of the related irrational sequences. The main results are two criteria characterising lower bounds for the irrationality measures of certain sequences. Applications and several examples are included.

Full text

Hiroshima Math. J.
35 (2005), 183–195
Irrationality measure of sequences*
Jaroslav Hanc
ˇland Ferdina
´nd Filip
(Received November 28, 2003)
(Revised September 29, 2004)
Abstract.The new concept of an irrationality measure of sequences is introduced
in this paper by means of the related irrational sequences. The main results are two
criteria characterising lower bounds for the irrationality measures of certain sequences.
Applications and several examples are included.
1. Introduction
The concept of irrationality is very important in Diophantine approx-
imations. There are several criteria for the irrationality of numbers, see for
example, Erdo
¨s and Strauss [6], [7], Hanc
ˇl and Rucki [14], Borwein [1], [2] or
Borwein and Zhou [3]. Some interesting results concerning the Cantor series
can be found in the paper of Tijdeman and Pingzhi Yuan [17]. Let us mention
the book of Nishioka [16] which contains a nice survey of Mahler theory
including many results on irrationality. If we want to approximate a real
number by rationals then it is appropriate to introduce the so-called irratio-
nality measure of numbers.
Definition 1. Let xbe an irrational number. Then the number
lim sup
q!y
qAN
logqmin
pANxp
q


1
is called the irrationality measure of the number x.
Let us note that for such a measure we have the following theorem.
Theorem 1. Any irrational number has an irrationality measure greater or
equal to 2.
The proof of Theorem 1 can be found in the book of Hardy and Wright in
[15]. The result concerning the lower bound for the irrationality measure of
* Supported by the grants no. 201/04/0381 and MSM6198898701
2000 Mathematics Subject Classification. 11J82.
Key words and phrases. Sequences, Irrationality, Irrationality measure.

the sum of infinite series which consist of terms of rational numbers is included
in the paper of Duverney [4] for instance. In 1975 Erdo
¨s [5] defined irrational
sequences in the following way.
Definition 2. Let fangy
n¼1be a sequence of positive real numbers. If for
every sequence fcngy
n¼1of positive integers the sum of the series
X
y
n¼1
1
ancn
is an irrational number, then the sequence fangy
n¼1is called irrational.If
fangy
n¼1is not an irrational sequence, then it is a rational sequence.
Erdo
¨s [5] also proved that the sequence f22ngy
n¼1is irrational. Some
generalizations and similar criteria can be found in [8], [9], [11] or [12]. To
each irrational sequence fangy
n¼1we can associate the sums of infinite series
Py
n¼11
cnan;cnAN
no
which are all irrational numbers. If we want to ap-
proximate such a set by rationals then it is suitable to introduce the so-called
irrationality measure of sequences in the following way.
Definition 3. Let fangy
n¼1be an irrational sequence. Let Cbe the set of
all sequences of positive integers, C¼ffcngy
n¼1;cnANg. Then the number
inf
fcngy
n¼1AClim sup
q!y
qAN
logqmin
pANX
y
n¼1
1
ancn
p
q

!
1
is called the irrationality measure of the sequence fangy
n¼1.
Unfortunately it is impossible to find a version of Duverney’s criterion (see [4])
for irrationality measure in the case of irrational sequences. We now introduce
Theorem 2 and Theorem 3 which are new criteria.
2. Main result
Theorem 2. Let e,e1and S be three positive real numbers such that
e1<e
1þeð1Þ
and
S>1
1e1
:ð2Þ
184 Jaroslav Hanc
ˇland Ferdina
´nd Filip

Assume that fangy
n¼1and fbngy
n¼1are two sequences of positive integers such that
fangy
n¼1is nondecreasing, and that
lim sup
n!y
a1=ðSþ1Þn
n>1;ð3Þ
bn¼Oðae1
nÞ;ð4Þ
and for every su‰ciently large positive integer n
an>n1þe:ð5Þ
Then the sequence an
bn
no
y
n¼1is irrational and has the irrationality measure greater
than or equal to maxð2;Sð1e1ÞÞ.
Theorem 3. Let eand S be two positive real numbers with S >1.
Assume that fangy
n¼1and fbngy
n¼1are two sequences of positive integers, such that
fangy
n¼1is nondecreasing, (3) and (5) for every su‰ciently large positive integer n
hold, and that for every positive real number b
bn¼oðab
nÞ:ð6Þ
Then the sequence an
bn
no
y
n¼1is irrational and has the irrationality measure greater
than or equal to maxð2;SÞ.
3. Proofs
Lemma 1. Let e1be a positive real number such that e1<1. Assume that
fangy
n¼1and fbngy
n¼1are two sequences of positive integers with fangy
n¼1non-
decreasing, such that
bn¼Oðae1
nÞð7Þ
and
an>2nð8Þ
for every su‰ciently large n.Then for every e2with e2>e1and su‰ciently
large n
X
y
j¼0
bnþj
anþj
<1
a1e2
n
:ð9Þ
Proof (of Lemma 1). Let nbe a su‰ciently large positive integer such
that (8) holds. From equation (7) we obtain that there exists a positive real
number Kwhich does not depend on nand such that
185Irrationality measure of sequences

bnaKae1
n:ð10Þ
Inequality (10) implies that
X
y
j¼0
bnþj
anþj
aX
y
j¼0
Kae1
nþj
anþj
¼X
y
j¼0
K
a1e1
nþj
¼X
nanþj<log2an
K
a1e1
nþj
þX
nþjblog2an
K
a1e1
nþj
:ð11Þ
Now we will estimate the both sums on the right hand side of inequality (11).
For the first sum of (11), we obtain that
X
nanþj<log2an
K
a1e1
nþj
aKlog2an
a1e1
n
:ð12Þ
For the second sum of (11), inequality (8) yields
X
nþjblog2an
K
a1e1
nþj
aX
nþjblog2an
K
2ðnþjÞð1e1Þ
aK
að1e1Þ
nX
y
j¼0
1
2jð1e1ÞaL
a1e1
n
;ð13Þ
where Lis a suitable positive real constant which does not depend on n. From
(11), (12) and (13) we obtain that for every e2with e2>e1and for every
su‰ciently large positive integer n
X
y
j¼0
bnþj
anþj
aX
nanþj<log2an
K
a1e1
nþj
þX
nþjblog2an
K
a1e1
nþj
aKlog2anþL
a1e1
n
a1
a1e2
n
:
Thus (9) holds. The proof of Lemma 1 is complete. r
Lemma 2. Let S, e1,fangy
n¼1and fbngy
n¼1satisfy all conditions in Theorem
2. Then there exists a positive real number asuch that for every su‰ciently
large n
X
y
j¼0
bnþj
anþj
a1
aa
n
:ð14Þ
186 Jaroslav Hanc
ˇland Ferdina
´nd Filip

Proof (of Lemma 2). From (4) we obtain that there exists a positive real
constant K, such that
bnaKae1
n:ð15Þ
Inequality (15) implies
X
y
j¼0
bnþj
anþj
aX
y
j¼0
Kae1
nþj
anþj
¼X
y
j¼0
K
a1e1
nþj
¼X
nanþj<a1=ð1þeÞ
n
K
a1e1
nþj
þX
nþjba1=ð1þeÞ
n
K
a1e1
nþj
:ð16Þ
Now we will estimate the both sums on the right hand side of inequality (16).
For the first sum, we obtain that
X
nanþj<a1=ð1þeÞ
n
K
a1e1
nþj
aKa1=ð1þeÞ
n
a1e1
n
¼K
a11=ð1þeÞe1
n
¼K
ae=ð1þeÞe1
n
:ð17Þ
For the second sum, inequality (5) implies that there exist positive real
constants Vand Rnot depending on n, such that
X
nþjba1=ð1þeÞ
n
K
a1e1
nþj
aX
nþjba1=ð1þeÞ
n
K
ðnþjÞð1þeÞð1e1Þ
aðy
a1=ð1þeÞ
n
Vdx
xð1þeÞð1e1ÞaR
ae=ð1þeÞe1
n
:ð18Þ
From (16), (17) and (18) we obtain that
X
y
j¼0
bnþj
anþj
aX
nanþj<a1=ð1þeÞ
n
K
a1e1
nþj
þX
nþjba1=ð1þeÞ
n
K
a1e1
nþj
aK
ae=ð1þeÞe1
n
þR
ae=ð1þeÞe1
n
¼KþR
ae=ð1þeÞe1
n
:ð19Þ
Let a¼1
2
e
1þee1

. Then inequality (1) implies that a>0. This and (19)
yield that for every su‰ciently large n
X
y
j¼0
bnþj
anþj
a1
aa
n
:
Thus (14) holds. The proof of Lemma 2 is complete. r
187Irrationality measure of sequences

Proof (of Theorem 2). Let fcngy
n¼1be a sequence of positive integers.
Then there exists a bijection f:N!Nsuch that for every nAN,An¼
afðnÞcfðnÞand the sequence fAngy
n¼1is nondrecreasing. From this description of
a bijection fand from the fact that fangy
n¼1is a nondecreasing sequence of
positive integers we obtain that for every nAN
An¼afðnÞcfðnÞban:ð20Þ
Let Bndenote bfðnÞfor all nAN. This and (20) imply that the sequences
fAngy
n¼1and fBngy
n¼1satisfy all asumptions of Theorem 2 too. From this fact,
(2) and Theorem 1 we obtain that to prove Theorem 2 it is su‰cient to prove
that for every real number S1with
S1>e1S;ð21Þ
and with S1e1Ssu‰ciently small we have
lim inf
N!yY
N1
n¼1
an
!
SS1
X
y
n¼N
bn
an
¼0:ð22Þ
Inequality (21) implies that there is a positive real number e2with e2>e1and
such that S1>e2S. Let us put
d¼S1e2S
2:
Then d>0 and we have
e2þSS1
Sd¼1S1e2Sdþe2d
Sd
¼1
S1e2S
2þe2d
Sd<1:ð23Þ
Inequality (3) implies that
lim sup
n!y
a1=ðSþ1dÞn
n¼y:ð24Þ
Let Abe a su‰ciently large positive real number. From (24) we obtain that
there exists a positive integer nsuch that
a1=ðSþ1dÞn
n>A:
Assume that ais a positive real number which satisfies condition (14) in
Lemma 2. Let for every positive integer kbmaxða1;3Þ,wkdenote the least
positive integer such that
188 Jaroslav Hanc
ˇland Ferdina
´nd Filip

a1=ðSþ1dÞwk
wk>k2=a:ð25Þ
Suppose that tkis the greatest positive integer less than wksuch that
atkaktk:ð26Þ
Let vkbe the least positive integer greater than tksuch that
a1=ðSþ1dÞvk
vk>k:ð27Þ
From the description of sequences ftkgy
k¼a1,fvkgy
k¼a1,fwkgy
k¼a1and from (25),
(26) and (27) we obtain that
tk<vkawk;ð28Þ
lim
k!yvk¼y
and for every positive integers rand swith vkarawkand tk<s<vk
ar>kr>2rð29Þ
and
as<kðSþ1dÞs:ð30Þ
The fact that the sequence fangy
n¼1of positive integers is nondecreasing and
inequality (26) imply that
Y
tk
n¼1
anaatk
tkakt2
k:ð31Þ
From (28) and (30) we obtain
kðSþ1dÞvkbkðSdÞððSþ1dÞvk1þðSþ1dÞvk2þþ1Þ
¼Y
vk1
n¼1
kðSþ1dÞn
!
ðSdÞ
bY
vk1
n¼1
an
!
ðSdÞ
Y
tk
n¼1
an
!
ðSdÞ
:
This fact, (28) and (31) yield
kðSþ1dÞvkbY
vk1
n¼1
an
!
ðSdÞ
Y
tk
n¼1
an
!
ðSdÞ
bY
vk1
n¼1
an
!
ðSdÞ
kt2
kðSdÞbY
vk1
n¼1
an
!
ðSdÞ
kv2
kðSdÞ:ð32Þ
189Irrationality measure of sequences

Inequality (32) implies that
Y
vk1
n¼1
an
!
ðSS1Þ
akððSS1Þ=ðSdÞÞðSþ1dÞvkþðSS1Þv2
k:ð33Þ
From (27), (29) and Lemma 1 we obtain the fact that
X
wk
n¼vk
bn
an
a1
a1e2
vk
a1
kð1e2ÞðSþ1dÞvk:ð34Þ
Inequality (25), the fact that the sequence fangy
n¼1is nondecreasing and Lemma
2 yield
X
y
n¼wkþ1
bn
an
a1
aa
wkþ1
a1
aa
wk
a1
k2ðSþ1dÞwk:ð35Þ
Since 2 >1e2then, (28), (34) and (35) imply that for every su‰ciently
large k
X
y
n¼vk
bn
an
¼X
wk
n¼vk
bn
an
þX
y
n¼wkþ1
bn
an
a1
kð1e2ÞðSþ1dÞvkþ1
k2ðSþ1dÞwka2
kð1e2ÞðSþ1dÞvk:ð36Þ
From (33) and (36) we obtain that for every su‰ciently large vk
Y
vk1
n¼1
an
!
ðSS1Þ
X
y
n¼vk
bn
an
akðSS1Þv2
kð1ðe2þðSS1Þ=ðSdÞÞÞðSþ1dÞvk:ð37Þ
Inequality (23) yields that 1 e2þSS1
Sd

>0. From this fact and (37) we
obtain the fact that
lim
k!yY
vk1
n¼1
an
!
SS1
X
y
n¼vk
bn
an
¼0:
This implies (22). The proof of Theorem 2 is now complete. r
Proof (of Theorem 3). Suppose that the sequence an
bn
no
y
n¼1has an ir-
rationality measure less than S. Then there exists a positive real number
S1<min S1;e
1þe

such that the irrationality measure of the sequence
an
bn
no
y
n¼1is less than SS1. Let e1¼S1
S. Then (6) implies that bn¼oðae1
nÞ.
Hence bn¼Oðae1
nÞ. From this and Theorem 2 we obtain that the sequence
190 Jaroslav Hanc
ˇland Ferdina
´nd Filip

an
bn
no
y
n¼1is irrational and has irrationality measure greater than or equal to
maxð2;Sð1e1ÞÞ. This is a contradiction since Sð1e1Þ¼SSe1¼SS1.
r
4. Examples and comments
Corollary 1. Let e1and S be positive real numbers such that
Sð1e1Þ>2. Assume that fangy
n¼1and fbngy
n¼1are two sequences of positive
integers such that fangy
n¼1is nondecreasing, with
lim sup
n!y
a1=ðSþ1Þn
n>1
and
bn¼Oðae1
nÞ:
Then the sequence an
bn
no
y
n¼1has irrationality measure greater than or equal to
Sð1e1Þ.
Corollary 1 is an immediate consequence of Theorem 2.
Example 1. Corollary 1 implies that the sequence
35nþn5
25nþn5

y
n¼1
has irrationality measure greater than or equal to 4ðlog231Þ
log23.
Example 2. Let Kbe a positive integer with K11
log2e

>2. Denote
that lcmðx1;x2;...;xnÞis the least common multiple of the numbers
x1;x2;...;xn. Then Corollary 1 yields that irrationality measure of the
sequence
lcmð1;2;...;ðKþ1ÞnÞþn
2ðKþ1Þnþn2

y
n¼1
is greater than or equal to K11
log2e

.
Corollary 2. Let S be a positive real number with S >2. Assume that
fangy
n¼1and fbngy
n¼1are two sequences of positive integers, that fangy
n¼1is
nondecreasing, that
lim sup
n!y
a1=ðSþ1Þn
n>1
with
191Irrationality measure of sequences

bn¼oðab
nÞ
for every positive real number b.Then the sequence an
bn
no
y
n¼1has irrationality
measure greater than or equal to S.
Corollary 2 is an immediate consequence of Theorem 3.
Example 3. Corollary 2 yields that the sequence
34nþ2n
33nþ5n

y
n¼1
has irrationality measure greater than or equal to 3.
Example 4. Let Sbe a positive real number with Sb2. Assume that
pðxÞis the number of primes less than or equal to x. As an immediate
consequence of Corollary 2 we obtain that the sequence
fpððSþ1ÞnÞ!þ1gy
n¼1
has irrational measure greater than or equal to S.
Example 5. Let Kbe a positive integer such that K>3. Also let ½xbe
the greatest integer less than or equal to x. Then Theorem 2 implies that the
sequence
2nþ3ðKþ1Þ22½log2log 2n
þn2
21þðKþ1Þ22½log2log 2n
þn
8
<
:9
=
;
y
n¼1
has irrationality measure greater than or equal to 2K
3.
Example 6. Let Kbe a positive real number such that K>2. Then
Theorem 3 yields that the sequence
2nþðKþ1Þ22½log2log 2n
þn!
2pððKþ1Þ22½log2log 2n
Þþnn
8
<
:9
=
;
y
n¼1
has irrationality measure greater than or equal to K.
Definition 4. Let xbe an irrational number. If the irrationality
measure of the number xis infinity then xis called Liouville number. Let
fangy
n¼1be a sequence of positive real numbers. If for every sequence fcngy
n¼1
of positive integers, the sum of the series Py
n¼11
ancnis a Liouville number, then
the sequence fangy
n¼1is called Liouville.
192 Jaroslav Hanc
ˇland Ferdina
´nd Filip

Corollary 3. Let eand e1be two positive real numbers with e1<e
1þe.
Assume that fangy
n¼1and fbngy
n¼1are two sequences of positive integers, that
fangy
n¼1is nondecreasing, and that
bn¼Oðae1
nÞ;
with, for every positive real number S,
lim sup
n!y
a1=Sn
n¼y
and for every su‰ciently large positive integer n, an>n1þe.Then the sequence
an
bn
no
y
n¼1is Liouville.
Corollary 3 is an immediate consequence of Theorem 2 and Definition 4.
Example 7. As an immediate consequence of Corollary 3 we obtain that
the sequences
22n!þn
2n!þn!

y
n¼1
and 23nnþ1
2nnþ1

y
n¼1
are Liouville.
Corollary 4. Let ebe a positive real number. Assume that fangy
n¼1and
fbngy
n¼1are two sequences of positive integers. Suppose that fangy
n¼1is non-
decreasing, and that for every positive real number b
bn¼oðab
nÞ:
Finally assume that for every positive real number S,
lim sup
n!y
a1=Sn
n¼y
and for every su‰ciently large positive integer n, an>n1þe.Then the sequence
an
bn
no
y
n¼1is Liouville.
Corollary 4 is an immediate consequence of Corollary 3.
Example 8. As an immediate consequence of Corollary 4 we obtain that
the sequences
nn!þ1
2n!þ1

y
n¼1
and 2ðnþ1Þ!þ1
2n!þ1

y
n¼1
are Liouville.
193Irrationality measure of sequences

Remark 1. In either Corollary 3or Corollary 4choose bn¼1for every
nAN.Then we obtain Erdo
¨s theorem which states the following. Let fangy
n¼1
be a sequence of positive integers such that for every positive integer S,
lim sup
n!y
a1=Sn
n¼y:
Let also ebe a positive real number such that for every su‰ciently large positive
integer n, an>n1þe.Then the sum of the series Py
n¼11
anis a Liouville number.
For more details see [5] or [10], for instance.
Open Problem. We do not know if the sequence f44ngy
n¼1has the ir-
rationality measure greater than 3.
Acknowledgement
We would like thank Professor Radhakrishnan Nair of the Department of
Mathematical Sciences, Liverpool University for his help with this article.
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194 Jaroslav Hanc
ˇland Ferdina
´nd Filip

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Jaroslav Hanc
ˇl
Institute for Research and Applications of Fuzzy Modeling and
Department of Mathematics
University of Ostrava, 30. dubna 22
701 03, Ostrava 1, Czech republic
e-mail: hancl@osu.cz
Ferdina
´nd Filip
Institute for Research and Applications of Fuzzy Modeling and
Department of Mathematics
University of Ostrava, 30. dubna 22
701 03, Ostrava 1, Czech republic
e-mail: filip.ferdinand@seznam.cz
195Irrationality measure of sequences