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Hiroshima Math. J.
35 (2005), 183–195
Irrationality measure of sequences*
Jaroslav Hančl and Ferdinánd 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 approximations. There are several criteria for the irrationality of numbers, see for
example, Erdös and Strauss [6], [7], Hančl 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 irrationality measure of numbers.
Definition 1.
Let x be an irrational number.
Then the number
p 1
lim sup log q minx pAN
q
q!y
qAN
is called the irrationality measure of the number x.
Let us note that for such a measure we have the following theorem.
Theorem 1.
equal to 2.
Any irrational number has an irrationality measure greater or
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.
184
Jaroslav Hančl and Ferdinánd Filip
the sum of infinite series which consist of terms of rational numbers is included
in the paper of Duverney [4] for instance. In 1975 Erdös [5] defined irrational
sequences in the following way.
Definition 2. Let fa n gy
n¼1 be a sequence of positive real numbers.
every sequence fcn gy
of
positive
integers the sum of the series
n¼1
If for
y
X
1
a c
n¼1 n n
is an irrational number, then the sequence fa n gy
n¼1 is called irrational. If
y
fa n gn¼1 is not an irrational sequence, then it is a rational sequence.
n
Erdös [5] also proved that the sequence f2 2 gy
n¼1 is irrational. Some
generalizations and similar criteria can be found in [8], [9], [11] or [12]. To
each irrational sequence
fa n gy
n¼1 we can associate the sums of infinite series
n
o
Py 1
which are all irrational numbers. If we want to apn¼1 cn a n ; cn A N
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 fa n gy
n¼1 be an irrational sequence. Let C be the set of
all sequences of positive integers, C ¼ ffcn gy
n¼1 ; cn A Ng. Then the number
inf
fcn gy
n¼1 A C
!1
X
y
1
p
min
p A N
a c
q
n¼1 n n
lim sup logq
q!y
qAN
is called the irrationality measure of the sequence fa n gy
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, e1 and S be three positive real numbers such that
e
1þe
ð1Þ
1
:
1 e1
ð2Þ
e1 <
and
S>
Irrationality measure of sequences
185
y
Assume that fa n gy
n¼1 and fbn gn¼1 are two sequences of positive integers such that
fa n gy
n¼1 is nondecreasing, and that
n
> 1;
lim sup a 1=ðSþ1Þ
n
ð3Þ
bn ¼ Oða ne1 Þ;
ð4Þ
n!y
and for every su‰ciently large positive integer n
a n > n 1þe :
Then the sequence
n oy
an
bn
n¼1
ð5Þ
is irrational and has the irrationality measure greater
than or equal to maxð2; Sð1 e1 ÞÞ.
Theorem 3. Let e and S be two positive real numbers with S > 1.
y
Assume that fa n gy
n¼1 and fbn gn¼1 are two sequences of positive integers, such that
y
fa n gn¼1 is nondecreasing, (3) and (5) for every su‰ciently large positive integer n
hold, and that for every positive real number b
n oy
bn ¼ oða nb Þ:
ð6Þ
a
Then the sequence bnn
is irrational and has the irrationality measure greater
n¼1
than or equal to maxð2; SÞ.
3. Proofs
Lemma 1. Let e1 be a positive real number such that e1 < 1. Assume that
y
y
fa n gy
n¼1 and fbn gn¼1 are two sequences of positive integers with fa n gn¼1 nondecreasing, such that
bn ¼ Oða ne1 Þ
ð7Þ
an > 2 n
ð8Þ
and
for every su‰ciently large n.
large n
Then for every e2 with e2 > e1 and su‰ciently
y
X
bnþj
j¼0
anþj
<
1
2
a 1e
n
:
ð9Þ
Proof (of Lemma 1). Let n be a su‰ciently large positive integer such
that (8) holds. From equation (7) we obtain that there exists a positive real
number K which does not depend on n and such that
186
Jaroslav Hančl and Ferdinánd Filip
bn a Ka ne1 :
ð10Þ
Inequality (10) implies that
y
X
bnþj
j¼0
anþj
a
y Ka e1
X
nþj
j¼0
¼
¼
anþj
y
X
K
1e1
anþj
j¼0
X
K
þ
1e1
nanþj<log 2 a n anþj
X
K
1e1
nþjblog 2 a n anþj
:
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
K
1e1
nanþj<log 2 a n anþj
K log 2 a n
a
1
a 1e
n
ð12Þ
:
For the second sum of (11), inequality (8) yields
X
K
1e1
nþjblog 2 a n anþj
X
a
nþjblog 2 a n
a
K
K
2ðnþjÞð1e1 Þ
y
X
ð1e Þ
a n 1 j¼0
1
2 jð1e1 Þ
a
L
1
a 1e
n
;
ð13Þ
where L is a suitable positive real constant which does not depend on n. From
(11), (12) and (13) we obtain that for every e2 with e2 > e1 and for every
su‰ciently large positive integer n
y
X
bnþj
j¼0
anþj
a
a
Thus (9) holds.
X
K
1e1
nanþj<log 2 a n anþj
K log 2 a n þ L
1
a 1e
n
þ
a
X
K
1e1
nþjblog 2 a n anþj
1
2
a 1e
n
:
The proof of Lemma 1 is complete.
r
y
Lemma 2. Let S, e1 , fa n gy
n¼1 and fbn gn¼1 satisfy all conditions in Theorem
2. Then there exists a positive real number a such that for every su‰ciently
large n
y
X
bnþj
j¼0
anþj
a
1
:
a na
ð14Þ
Irrationality measure of sequences
Proof (of Lemma 2).
constant K, such that
187
From (4) we obtain that there exists a positive real
bn a Ka ne1 :
ð15Þ
Inequality (15) implies
y
X
bnþj
j¼0
a
anþj
y Ka e1
X
nþj
j¼0
anþj
y
X
K
¼
X
¼
1e1
anþj
j¼0
K
1=ð1þeÞ
nanþj<a n
1e1
anþj
X
þ
K
1=ð1þeÞ
nþjba n
ð16Þ
:
1e1
anþj
Now we will estimate the both sums on the right hand side of inequality (16).
For the first sum, we obtain that
X
1=ð1þeÞ
K
a
1e1
1=ð1þeÞ anþj
Ka n
¼
1
a 1e
n
nanþj<a n
K
11=ð1þeÞe1
an
¼
K
e=ð1þeÞe1
an
:
ð17Þ
For the second sum, inequality (5) implies that there exist positive real
constants V and R not depending on n, such that
X
K
1=ð1þeÞ
nþjba n
1e1
anþj
X
a
K
1=ð1þeÞ
nþjba n
ðy
a
ðn þ jÞð1þeÞð1e1 Þ
V dx
1=ð1þeÞ
an
xð1þeÞð1e1 Þ
a
R
e=ð1þeÞe1
an
:
ð18Þ
From (16), (17) and (18) we obtain that
y
X
bnþj
j¼0
anþj
a
e
1þe
e1 .
K
1=ð1þeÞ
nanþj<a n
a
Let a ¼ 12
X
K
e=ð1þeÞe1
an
1e1
anþj
þ
X
þ
K
1e1
1=ð1þeÞ anþj
nþjba n
R
e=ð1þeÞe1
an
¼
K þR
e=ð1þeÞe1
an
:
ð19Þ
Then inequality (1) implies that a > 0. This and (19)
yield that for every su‰ciently large n
y
X
bnþj
j¼0
Thus (14) holds.
anþj
a
1
:
a na
The proof of Lemma 2 is complete.
r
188
Jaroslav Hančl and Ferdinánd Filip
Proof (of Theorem 2). Let fcn gy
n¼1 be a sequence of positive integers.
Then there exists a bijection f : N ! N such that for every n A N, An ¼
afðnÞ cfðnÞ and the sequence fAn gy
n¼1 is nondrecreasing. From this description of
a bijection f and from the fact that fa n gy
n¼1 is a nondecreasing sequence of
positive integers we obtain that for every n A N
An ¼ afðnÞ cfðnÞ b a n :
ð20Þ
Let Bn denote bfðnÞ for all n A N. This and (20) imply that the sequences
y
fAn gy
n¼1 and fBn gn¼1 satisfy 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 S1 with
ð21Þ
S1 > e1 S;
and with S1 e1 S su‰ciently small we have
!SS1
N1
y
Y
X
bn
an
¼ 0:
lim inf
N!y
a
n¼N n
n¼1
ð22Þ
Inequality (21) implies that there is a positive real number e2 with e2 > e1 and
such that S1 > e2 S. Let us put
d¼
S 1 e2 S
:
2
Then d > 0 and we have
e2 þ
S S1
S 1 e2 S d þ e2 d
¼1
Sd
Sd
¼1
S1 e2 S
2
þ e2 d
Sd
< 1:
ð23Þ
Inequality (3) implies that
n
lim sup a 1=ðSþ1dÞ
¼ y:
n
ð24Þ
n!y
Let A be a su‰ciently large positive real number. From (24) we obtain that
there exists a positive integer n such that
n
a1=ðSþ1dÞ
> A:
n
Assume that a is a positive real number which satisfies condition (14) in
Lemma 2. Let for every positive integer k b maxða1 ; 3Þ, wk denote the least
positive integer such that
Irrationality measure of sequences
a1=ðSþ1dÞ
wk
wk
189
> k 2=a :
ð25Þ
Suppose that t k is the greatest positive integer less than wk such that
at k a k t k :
ð26Þ
Let vk be the least positive integer greater than t k such that
vk
a1=ðSþ1dÞ
> k:
vk
ð27Þ
y
y
From the description of sequences ft k gy
k¼a1 , fvk gk¼a1 , fwk gk¼a1 and from (25),
(26) and (27) we obtain that
ð28Þ
t k < vk a w k ;
lim vk ¼ y
k!y
and for every positive integers r and s with vk a r a wk and t k < s < vk
ar > k r > 2 r
ð29Þ
and
s
as < k ðSþ1dÞ :
ð30Þ
The fact that the sequence fa n gy
n¼1 of positive integers is nondecreasing and
inequality (26) imply that
tk
Y
2
a n a attkk a k t k :
ð31Þ
n¼1
From (28) and (30) we obtain
vk
k ðSþ1dÞ b k ðSdÞððSþ1dÞ
¼
vY
k 1
k
vk 1
ðSþ1dÞ n
þðSþ1dÞ vk 2 þþ1Þ
!ðSdÞ
vY
k 1
b
n¼1
!ðSdÞ
an
tk
Y
!ðSdÞ
an
:
n¼1
n¼1
This fact, (28) and (31) yield
vk
k ðSþ1dÞ b
vY
k 1
!ðSdÞ
tk
Y
an
b
n¼1
an
n¼1
n¼1
vY
k 1
!ðSdÞ
!ðSdÞ
an
k
t 2k ðSdÞ
b
vY
k 1
n¼1
!ðSdÞ
an
2
k vk ðSdÞ :
ð32Þ
190
Jaroslav Hančl and Ferdinánd Filip
Inequality (32) implies that
!ðSS1 Þ
vY
k 1
vk
2
an
a k ððSS1 Þ=ðSdÞÞðSþ1dÞ þðSS1 Þvk :
ð33Þ
n¼1
From (27), (29) and Lemma 1 we obtain the fact that
wk
X
bn
n¼vk
an
a
1
2
av1e
k
a
1
v
k ð1e2 ÞðSþ1dÞ k
ð34Þ
:
Inequality (25), the fact that the sequence fa n gy
n¼1 is nondecreasing and Lemma
2 yield
y
X
bn
1
1
1
a a
a a a 2ðSþ1dÞ wk :
a
a
a
k
wk
wk þ1
n¼wk þ1 n
ð35Þ
Since 2 > 1 e2 then, (28), (34) and (35) imply that for every su‰ciently
large k
y
X
bn
n¼vk
an
¼
wk
X
bn
n¼vk
a
an
þ
y
X
bn
a
n¼w þ1 n
k
1
v
k ð1e2 ÞðSþ1dÞ k
þ
1
w
k 2ðSþ1dÞ k
a
2
v
k ð1e2 ÞðSþ1dÞ k
:
ð36Þ
From (33) and (36) we obtain that for every su‰ciently large vk
!ðSS1 Þ
vY
y
k 1
X
vk
bn
2
an
a k ðSS1 Þvk ð1ðe2 þðSS1 Þ=ðSdÞÞÞðSþ1dÞ :
ð37Þ
a
n¼vk n
n¼1
SS
Inequality (23) yields that 1 e2 þ Sd1 > 0. From this fact and (37) we
obtain the fact that
!SS1
vY
y
k 1
X
bn
an
¼ 0:
lim
k!y
a
n¼vk n
n¼1
This implies (22).
The proof of Theorem 2 is now complete.
n oy
a
Proof (of Theorem 3). Suppose that the sequence bnn
n¼1
r
has an ir-
rationality measure less than S. Then there exists a positive real number
e
S1 <
n
o min S 1; 1þe such that the irrationality measure of the sequence
an
bn
y
S
is less than S S1 . Let e1 ¼ S1 . Then (6) implies that bn ¼ oða ne1 Þ.
Hence bn ¼ Oða ne1 Þ. From this and Theorem 2 we obtain that the sequence
n¼1
Irrationality measure of sequences
n oy
an
bn
n¼1
191
is irrational and has irrationality measure greater than or equal to
maxð2; Sð1 e1 ÞÞ.
This is a contradiction since Sð1 e1 Þ ¼ S Se1 ¼ S S1 .
r
4. Examples and comments
Corollary 1. Let e1 and S be positive real numbers such that
y
Sð1 e1 Þ > 2. Assume that fa n gy
n¼1 and fbn gn¼1 are two sequences of positive
y
integers such that fa n gn¼1 is nondecreasing, with
n
>1
lim sup a 1=ðSþ1Þ
n
n!y
and
Then the sequence
Sð1 e1 Þ.
n oy
an
bn
n¼1
bn ¼ Oða ne1 Þ:
has irrationality measure greater than or equal to
Corollary 1 is an immediate consequence of Theorem 2.
Example 1.
Corollary 1 implies that the sequence
5n
y
3 þ n5
2 5 n þ n 5 n¼1
has irrationality measure greater than or equal to
4ðlog 2 31Þ
.
log 2 3
Example 2. Let K be a positive integer with K 1 log1 e > 2.
2
Denote
that lcmðx1 ; x2 ; . . . ; xn Þ is the least common multiple of the numbers
x1 ; x2 ; . . . ; xn . Then Corollary 1 yields that irrationality measure of the
sequence
y
lcmð1; 2; . . . ; ðK þ 1Þ n Þ þ n
n
2ðKþ1Þ þ n 2
n¼1
1
is greater than or equal to K 1 log e .
2
Corollary 2. Let S be a positive real number with S > 2. Assume that
y
y
fa n gy
n¼1 and fbn gn¼1 are two sequences of positive integers, that fa n gn¼1 is
nondecreasing, that
n
lim sup a 1=ðSþ1Þ
>1
n
n!y
with
192
Jaroslav Hančl and Ferdinánd Filip
bn ¼ oða nb Þ
for every positive real number b. Then the sequence
measure greater than or equal to S.
n oy
an
bn
n¼1
has irrationality
Corollary 2 is an immediate consequence of Theorem 3.
Example 3.
Corollary 2 yields that the sequence
4n
y
3 þ 2n
3 3 n þ 5 n n¼1
has irrationality measure greater than or equal to 3.
Example 4. Let S be a positive real number with S b 2. Assume that
pðxÞ is the number of primes less than or equal to x. As an immediate
consequence of Corollary 2 we obtain that the sequence
fpððS þ 1Þ n Þ! þ 1gy
n¼1
has irrational measure greater than or equal to S.
Example 5. Let K be 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
8
9y
½log log n
<2 nþ3ðKþ1Þ 2 2 2 2 þ n 2 =
: 1þðKþ1Þ 2 2½log 2 log 2 n
;
2
þ n n¼1
has irrationality measure greater than or equal to
2K
3
.
Example 6. Let K be a positive real number such that K > 2.
Theorem 3 yields that the sequence
8
9y
½log log n
< 2 nþðKþ1Þ 2 2 2 2 þ n! =
: pððKþ1Þ 2 2½log 2 log 2 n Þ
;
þ n n n¼1
2
Then
has irrationality measure greater than or equal to K.
Definition 4. Let x be an irrational number. If the irrationality
measure of the number x is infinity then x is called Liouville number. Let
fa n gy
If for every sequence fcn gy
n¼1 be a sequence of positive real numbers.
n¼1
Py 1
of positive integers, the sum of the series n¼1 a n cn is a Liouville number, then
the sequence fa n gy
n¼1 is called Liouville.
Irrationality measure of sequences
193
e
Corollary 3. Let e and e1 be two positive real numbers with e1 < 1þe
.
y
y
Assume that fa n gn¼1 and fbn gn¼1 are two sequences of positive integers, that
fa n gy
n¼1 is nondecreasing, and that
bn ¼ Oða ne1 Þ;
with, for every positive real number S,
n
lim sup a 1=S
¼y
n
n!y
andofor every su‰ciently large positive integer n, a n > n 1þe . Then the sequence
n
y
an
is Liouville.
bn
n¼1
Corollary 3 is an immediate consequence of Theorem 2 and Definition 4.
Example 7.
the sequences
As an immediate consequence of Corollary 3 we obtain that
2 2n! þ n
2 n! þ n!
y
and
n¼1
n
2 3n þ 1
2nn þ 1
y
n¼1
are Liouville.
Corollary 4. Let e be a positive real number. Assume that fa n gy
n¼1 and
are two sequences of positive integers. Suppose that fa n gy
n¼1 is nondecreasing, and that for every positive real number b
fbn gy
n¼1
bn ¼ oða nb Þ:
Finally assume that for every positive real number S,
n
lim sup a 1=S
¼y
n
n!y
andofor every su‰ciently large positive integer n, a n > n 1þe . Then the sequence
n
y
an
is Liouville.
bn
n¼1
Corollary 4 is an immediate consequence of Corollary 3.
Example 8.
the sequences
As an immediate consequence of Corollary 4 we obtain that
are Liouville.
n n! þ 1
2 n! þ 1
y
and
n¼1
2ðnþ1Þ! þ 1
2 n! þ 1
y
n¼1
194
Jaroslav Hančl and Ferdinánd Filip
Remark 1. In either Corollary 3 or Corollary 4 choose bn ¼ 1 for every
n A N. Then we obtain Erdös theorem which states the following. Let fa n gy
n¼1
be a sequence of positive integers such that for every positive integer S,
n
¼ y:
lim sup a 1=S
n
n!y
Let also e be a positive real number such that for every su‰ciently large positive
P
1
integer n, a n > n 1þe . Then the sum of the series y
n¼1 a n is a Liouville number.
For more details see [5] or [10], for instance.
n
Open Problem. We do not know if the sequence f4 4 gy
n¼1 has the irrationality 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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P
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Jaroslav Hančl
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: [email protected]
Ferdinánd 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: fi[email protected]