FRACTAL SUBSERIES OF THE HARMONIC SERIES
GÁBOR KORVIN 1 ABSTRACT: We study the convergence of certain subseries of the harmonic series corresponding to increasing sequences of integers whose digits in a certain base are not uniformly distributed. We also discuss the case of irregular sequences, where the frequency distribution of some of the digits does not exist. Examples are given for irregular sequences where the corresponding harmonic subseries is convergent, or divergent, respectively. 2000 Mathematical Subject Classification: 11K36 Key words and phrases: Integer sequences, irregular sequences, counting function, distribution of digits, harmonic series, Shannon entropy.
1. INTRODUCTION ∞
1 all terms n =1 n
Kempner [1] observed that if we omit from the harmonic series H := ∑
corresponding to integers whose decimal representation contains at least once the digit 9, then the remaining "9-free" subseries H 10 ,9 (where 10 refers to the base, 9 is the missing digit) will be convergent, and bounded by 90. Hansberger's [2] Mathematical Gems gives a simple proof of this fact, and discusses the case of other missing digits. More recent papers [3-5] give better bounds, the best published result [5] being 20.2 ≤ H 10 ,9 ≤ 28.3 . In this paper we first we generalize Kempner's [1] observation (Section 2), and discuss the convergence properties of some fractal subseries of the harmonic series. Recall that a countable infinite set ξ i ∈ [0, ∞) of real numbers is called fractal with mass fractal dimension α , 0 ≤ α < 1 if their counting function
N ( x) := # (ξ i | ξ i ≤ x ) behaves for
x → ∞ as N ( x) ∝ x α [6, 7]. In this spirit an increasing
1
sequence {ν i }of nonnegative
Box 1157, Earth Sciences Department, King Fahd University, Dhahran 31261, Saudi Arabia. Email:
[email protected]
1
integers will be called a fractal sequence of dimension α if their counting function asymptotically behaves as N ( n) := # (ν i | ν i ≤ n ) ∝ n with some α , 0 ≤ α < 1 . α
Section 3 is devoted to a more general case. The main result of this part (Theorem 6 and its Corollary 1) studies the convergence of the subseries of the harmonic series corresponding to integers with nonuniformly distributed digits in their base-r expansion. The convergence will depend on the Shannon entropy of the digits' frequency distribution (if it exists!). Finally (in Section 4) examples will be constructed for increasing sequences of integers (called irregular sequences) for which the frequency distribution does not exist for certain digits. Surprisingly (see Theorem 7), for some irregular sequences the corresponding harmonic subseries is convergent, while in some other cases it is divergent. The concluding Section 5 of the paper contains a simple conjecture, and related topics from Number Theory are indicated. Two technical Lemmas are relegated to the Appendix.
2. SIMPLE GENERALIZATIONS OF KEMPNER'S "9-FREE" SEQUENCES Theorem 1. Let ν r ,d = {ν i } be the set of those integers which in their base-r expansion (r>2) do not contain a certain digit d, where d can be 0,1, …r-1. Then the counting function α=
of
{ν i }, Nν (n ) :=# (ν i | ν i
≤ n)
scales
for
n→∞
as
N ν (n ) ∝ n α
with
⎛ 1 ⎞ log (r − 1) ⎟⎟ . + O⎜⎜ log r ⎝ log n ⎠
Remark 1. If r > 1 , the digits can be denoted by the symbols σ 1 = 0, σ 2 = 1, L , σ 10 = 9 ; and by arbitrary symbols for σ 11 ,L, σ r −1 , as e.g. in the hexadecimal system one uses 0,1, L ,9, A, B, L , F .
2
Proof of Theorem 1: Case a) Let the missing digit be d=0. All integers ν such that r k −1 ≤ ν ≤ r k have k digits in base-r, that is the number of those ν´s which do not contain the
digit d is (r − 1)k . Letting n = r K : k k K ⎛ ⎞ K Nν (n ) = ∑ # ⎜ ∑1 ⎟ = ∑ (r − 1) = −1 + ∑ (r − 1) k =0 k =1 ⎝ r k −1 ≤ν < r k ⎠ k =1 K +1 K +1 : , ( r − 1) − 1 (r − 1) K = −1 + ∝ ∝ (r − 1) = r αK r −2 r−2 K
with α =
⎛ 1 ⎞ log (r − 1) ⎟⎟ . + O⎜⎜ log r ⎝ log n ⎠
Case b) If the missing digit is d≠ 0, there are (r − 2 )(r − 1)
k −1
possible ν values such that
r k −1 ≤ ν ≤ r k , because the first digit cannot be 0. Thus, for n = r K ,
( )
Nν r
K
k −1
K −1 ⎛ ⎞ = ∑ # ⎜⎜ ∑ 1 ⎟⎟ = (r − 2 )∑ (r − 1) k =1 ⎝ r k −1 ≤ν < r k ⎠ k =1 K
k
K −2
= (r − 2 )∑ (r − 1) ∝ (r − 1) ∝ r αK , K
k =0
again with α =
⎛ 1 ⎞ log (r − 1) ⎟⎟ . + O⎜⎜ log r ⎝ log n ⎠
ð
Definition Let us call ζ r ,d ( x ) := ∑ ν ∈{ν
r ,d
1 }ν
x
(x real, r>2) the generalized Euler zeta function
corresponding to the sequence {ν r ,d } . We have the following Theorem 2. ζ r ,d ( x ) := ∑ ν ∈{ν
r ,d
1 }ν
x
is convergent if x >
3
log (r − 1) log r
and divergent if x ≤
log (r − 1) . log r
Proof.
1
ζ r ,d ( x ) := ∑
ν ∈{ν r ,d }
ζ r ,d ( x ) := ∑
1
ν ∈{υ r ,d }
In both cases if x >
νx
ν
x
∞
=∑
1
∑
l =1 r l −1 ≤ν < r l
∞
=∑
1
∑
l =1 r l −1 ≤ν < r l
log (r − 1) log r
νx
ν
x
∞
(r − 1)
l =1
r (l −1) x
α and divergent if x§α..
Proof. First we prove that if x> α then ς {ν } (x ) is convergent. Indeed, ∞
1
i =1
ν ix
ζ {ν } (x ) = ∑
∞
=∑
∑
1
l =1 R l −1 ≤ν i < R l
ν ix
∞
a+∑ l =1
∞ N (R l −1 ) − N (R l ) N (R l −1 ) − N (R l ) > a + ∑ R lx R ( l +1 ) x l =1
lα
b ∞ R = a + x ∑ lx = ∞ if α ≥ x. R l =1 R
ð
If N(n) is not a power function itself but asymptotically tends to such a function, we have a similar result : Theorem 4. Let {ν } = {ν 1 < ν 2 < L} be a monotone increasing sequence of integers with a counting function satisfying N (n ) ∝ n α [1+ f (n )] , α ∈ (0,1],
lim f (n) = 0 . Then there exists n →∞
an ε 0 such that for any 0 < ε < ε 0 < 1 the generalized zeta function corresponding to the sequence,
ζ {ν } (x ) := ∑ i
Proof. Let
1
ν ix
, is convergent if x>α(1+ε) and divergent if x ≤ α (1 − ε ) .
f (n) < ε1 < 1 for n ≥ R1 and n α (1−ε ) > 2 for n ≥ R2 and ε < ε 2 < 1 . Let
R = max{R1 , R2 }; ε 0 = min (ε 1 , ε 2 ,1) . It is sufficient to study the convergence of the
5
truncated
1
∑ν
series
i >R
∞
1
∑ν i≥ R
x i
=∑
∑
l =1 R l ≤ν i < R l +1
1
ν ix
α (1 + ε ) − x < 0.
if
1
∑ν l>L
x l
∞
>∑ l =1
=∑
∑
l > L R l ≤ν i ≤ R l +1
∞
≤∑
.
First,
N {ν } (R l +1 ) R lx
i =1
1+ ε )
l +1
R lx
1
the
x > α (1 + ε ).
let
(R )(
∞
< const1 ⋅ ∑
x ≤ α (1 − ε )
if
R lα (1+ ε ) const 3 ⋅ x ( l +1) x R R l
∞
∑
R
α (1−ε ) l
R lx
l =1
N (λR l ) = λ
( [ ])
N (R l +1 ) − N (R l ) ∞ N (R ) R >∑ =∑ R (l +1) x R (l +1)x l =1 l =1
x i
Then
= const 2 ⋅ ∑
scaling
∞
1
ν
x i
α 1+ f R l
(
α 1+ f R l
)
implies
)
−1
ð
.
= ∞ if x ≤ α (1 − ε )
Remark 3. Theorems 3 and 4 can also be deduced from the following known Theorem A (of Krzyś, Powell and Šalát, [9, 10, 11]): Let {m1 < m2 < L}be a sequence of integers with ∞
counting function N(n). Then
1
∑m n =1
< ∞ iff
∞
N (n) α (1 − δ ) and
n (n+1)
f ( n +1)
∞
∑ i =1
1
ν
x
= 1+
δ
⎛ 1 ⎞ + O⎜ p ⎟ n ⎝n ⎠
with
p >1
∞
.
Then
∑ i =1
1
ν
x
1 and
divergent
with β > 1 then the series
∑u
k
∞
α
k =1
q ≤ 1.
for
∞
Applying
this
to
∑n
x
is convergent for [1+ f ( n ) ]− 2
we
have
n =1
α
[1+ f ( n ) ]− 2
n α [ (n +1) x
1+ f ( n +1) ]− 2
x
⎛ 1⎞ = ⎜1 + ⎟ ⎝ n⎠
2−
α x
α
⎫x ⎧ n f (n) α⎤ 1 ⎛ 1 ⎞ ⎡ = 1 + ⎢2 + (δ − 1) ⎥ ⋅ + O⎜ p ⎟ . ⎨ f ( n +1) ⎬ x⎦ n ⎝n ⎠ ⎣ ⎭ ⎩ (n + 1)
ð
3. SEQUENCES WITH NON-UNIFORMLY DISTRIBUTED DIGITS In this part, we shall generalize Theorems 1 and 2 for subseries of the harmonic series corresponding to integers with non uniformly distributed digits in their base-r expansion ( r ≥ 2 ). We denote by the symbols σ 0 , σ 1 ,K, σ r −1 the digits 0,1, L used in the base-r number system. Let Λ = (lij )i =i ,i ,L; j =0,1,L,r −1 be a matrix of infinitely many rows and r 1
columns, consisting of nonnegative integer elements. Assume that Λ has the following properties: r −1
(A)
∑l
= i for i = i1 , i2 ,L with i1 ≤ i2 ≤ L; lim i = ∞ , n
ij
j =0
(B)
n →∞
lij
lim i i →∞
(C) λ j ≠ 0
:= λ j exists for
for
all
j = 0,1, L , r − 1 , and
j = 0,1, L , r − 1.
Then, of course, we also have r −1
(D)
∑λ j =0
j
= 1.
Denote by ν Λ the set of all positive integers ν for which:
8
(E) If r i −1 ≤ ν < r i then the base-r expansion of ν contains the digits σ 0 ,σ 1 ,K,σ r −1 exactly l i 0 , li1 , L , l i ,r −1 times ( i = i1 , i2 ,L ), respectively;
(F) The most significant (i.e. left-most) digit of ν is not zero. r −1
Theorem 6.
ζ ν (x ) := ∑ ν
ν∈
Λ
Λ
1
ν
x
is convergent if x >
− ∑ λ j log λ j j =0
and divergent if
log r
r −1
x≤
− ∑ λ j log λ j j =0
log r
. r −1
Remark 4. As the maximum of − ∑ λ j log λ j , subject to the conditions that λ j ≥ 0 for j =0
j = 0,1, L , r − 1 and
Corollary 1.
r −1
∑λ j =0
The
λ0 = λ1 = L = λ r −1 =
j
= 1 , is log r , Theorem 6 has the following Corrollary:
Λ-subseries of the harmonic series, H
ν
= Λ
1
∑ν ν ν
∈
is divergent if
Λ
1 and it is convergent otherwise. r
To prove Theorem 6 we shall need the following Lemma: Lemma 1. Denote by Θ r the set of those integers which in their base-r expansion ( r ≥ 2 ) do not contain all symbols σ 0 = 0, σ 1 = 1, σ 2 ,K , σ r −1 at least once . Then the series
1
∑ ν is ν Θr
∈
convergent. Proof of Lemma 1.
As in Theorem 1, we denote by ν r ,σ = {ν } the set of those integers
which in their base-r expansion ( r ≥ 2 ) do not contain a certain digit σ, where σ can be any of
σ 0 = 0, σ 1 = 1, σ 2 ,K , σ r −1 . Also, as in Theorem 2, we use the notation
9
ζ
Θr
( x) :=
1
∑ν Θr
ν∈
x
. Then we have
1
∑ν
ν ∈Θ
Theorem 2 for all k = 0,1, L , r − 1
ζ
r ,σ k
r −1
N ( r , i ) if we also count such numbers ν ∈ν Λ* whose first digit is 0, that is if we
write N (r , i) < N (r , i) =
i! . Because of conditions B and C, there exists a li 0 !li1!Lli ,r −1!
sufficiently large I such that for i>I even n = min{l i 0 , L , l i ,r −1 } will be as large that we can ⎛n⎞ apply Stirling's formula n!~ ⎜ ⎟ ⎝e⎠
n
2πn . The formula gives
r −1 r −1 ⎤ 1 1 r −1 ⎡ ⎤ ⎡ r −1 log N (r , i ) < log i!−∑ log l ij ≈ ⎢i log i − i + log 2πi ⎥ − ⎢∑ l ij log l ij − ∑ l ij + ∑ log 2πl ij ⎥ 2 2 j =0 ⎣ ⎦ ⎣ j =0 j =0 j =0 ⎦ r −1
= −i ∑ j =0
lij i
log
r −1 l lij 1 1 1 r −1 ij + log 2πi − ∑ log 2πlij < −i ∑ log + log 2πi, i 2 i 2 2 j =0 j =0 i
lij
that is lij ⎤ ⎡ li ,r −1 ⎞⎤ ⎡ r −1 lij ⎛l ⎟⎥ N (r , i ) < N (r , i ) < 2πi exp ⎢− i ∑ log ⎥ = 2πi exp ⎢i ⋅ E ⎜⎜ i 0 , L , i⎦ i ⎟⎠⎦ ⎝ i ⎣ j =0 i ⎣
10
(1)
r −1 l l li ,r −1 ⎞ ⎛l ⎟⎟ = −∑ ij log ij where E ⎜⎜ i 0 ,L, i ⎠ i j =0 i ⎝ i
is the Shannon entropy of the probability
li ,r −1 ⎞ ⎛l ⎟ . By Condition B, and the continuity of the Shannon entropy, for distribution ⎜⎜ i 0 ,L, i ⎟⎠ ⎝ i an arbitrarily small positive ε there exists an iε
such that for i > iε
one has
li ,r −1 ⎞ l ⎛l ⎟⎟ ≤ (1 + ε ) E (λ0 , λ1 ,L, λr −1 ) where λ j = lim ij . For such an i we have E ⎜⎜ i 0 , L, i ⎠ i →∞ i ⎝ i
N (r , i) < 2πi exp[i ⋅ E (λ0 , L, λr −1 )] .
(2)
Introduce the notations i * = max{iε , r }, E (λ0 , λ1 , L, λr −1 ) = E , 2π = B . Then 1
∑ν ν
ν∈
*
∞
1
< A+∑
∞
∑ < A + ∑* i =i* ν ∈ * ν i =i νΛ i i −1 r ≤ν 2, there exist such irregular sequences ν Λ with the further property that conv
1
∑ν ν
ν∈
∗
< ∞ , and there exist irregular sequences ν Λ for which div
Λ
13
1
∑ν ν
ν∈
∗
Λ
= ∞.
Remark 5. The term "irregular" in the Theorem has been borrowed from papers of Barreira et al. [14, 15] who called a real number α ∈ (0,1) irregular or atypical if the frequency distribution of digits in their decimal (or in any other base r) expansion does not exist. Proof. To construct a simple irregular sequence ν Λ
conv
digits, say σ
j0
in base r, select any of the r possible
= σ and replace all other digits by the symbol ∗ . Consider the sequence
⎧ν 0 = 1 ⎪ ⎪ν 1 = 1σ ⎪ ⎪ν 2 = 1σ ∗ ⎪ = 1σ ∗ ∗ ⎪ν 3 ⎪ν = 1σ ∗ ∗σ ⎪ 4 ⎪ν 5 = 1σ ∗ ∗σσ ⎪ ⎪ν 6 = 1σ ∗ ∗σσσ ⎪ ⎪ν 7 = 1σ ∗ ∗σσσσ ⎨ ⎪ν 8 = 1σ ∗ ∗σσσσ ∗ ⎪ = 1σ ∗ ∗σσσσ ∗ ∗ ⎪ν 9 ⎪M ⎪ ⎪ν 15 = 1σ ∗ ∗σσσσ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎪ ⎪ν 16 = 1σ ∗ ∗σσσσ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗σ ⎪M ⎪ : ⎪ν 31 = 1σ L3 σ ∗ ∗σσσσ ∗ ∗ ∗43 ∗ ∗ ∗σ {{ 123∗1∗42 12 ⎪ 1× 2× 8× 4× 16× ⎪M ⎩ It is easy to check that
lim i →∞
number
lim i →∞
ν
i
liσ := λ σ (where i
l σ is the number of the i
) does not exist, because the limit is:
l iσ 1 2 k −1 = if i → ∞ as i = 1,7,31, L, 2 − 1, L (k = 1,2,3,L) i 3
14
σ -digits in the
lim i →∞
l iσ 2 2k = if i → ∞ as i = 3,15,63, L, 2 − 1, L (k = 1,2,3,L) . i 3
More generally, with the notation σ n = σσ σ , n ≥ 1, we can construct the sequence L 1 42 4 3 n − times
⎧ν 0 = γ ⎪ ⎪ν 1 = γσ ⎪ ⎪ν 2 = γσ ∗ ⎪ = γσ ∗ 2 ⎪ν 3 ⎪M ⎪ k ⎪ν k +1 = γσ ∗ ⎪ {ν } =ν (Λk ) = ⎨ν k + 2 = γσ ∗k σ ⎪ ⎪M 2 ⎪ 2 = γσ ∗ k σ k ν k k + + 1 ⎪ ⎪M ⎪ 2 3 ⎪ν k 3 + k 2 + k +1 = γσ ∗ k σ k ∗ k ⎪ ⎪M ⎪ ⎩
where ∗ ≠ σ , γ ≠ σ ,0 can be arbitrary digits. It is easy to check (see Lemma 2 in the Appendix) that
lim inf i →∞
lσ i
i
=
1 k , ; lim sup l iσ = k +1 i k +1 i →∞
(8)
⎧⎪ ⎫⎪ that is the sequence ⎨ l iσ ⎬ is not convergent if k > 1 . There are at least two convergent ⎪⎩ i ⎪⎭
subsequences of
⎧⎪ l iσ ⎫⎪ 1 k and . These ⎨ ⎬ which tend to different limits, namely to k +1 k +1 ⎪⎩ i ⎪⎭
15
are obtained when i → ∞ through the numbers
i=
i=
k 2m+ 2 − 1 , or through the numbers k −1
k 2m+ 4 − 1 , (m = 1,2,3,L) , respectively. k −1
(γ ≠ σ , γ
If r>2 and we substitute for all ∗-digits the value γ
sequence
ν
k ,conv Λ
. (The convergence of the corresponding series
ν
k ,conv
i →∞
1
k , onv
Λ
follows from
ν
⊆ Θ r .)
Next we construct a sequence
lim i
∑
ν∈
Lemma 1, because ν Λ
l ij
≠ 0) , we get an irregular
ν
k , div
for which
Λ
(when represented in base r,
:= λ j does not exist for some j ∈ {1,2, L, r − 1} , and the sum
∑
ν
ν∈
Starting out from the previous sequence
{ν }=ν
k , conv Λ
k , div Λ
1
ν
r>2)
is divergent.
we can include further numbers to it
if we select the ∗-digits less restrictively, for example by allowing each of them to be any symbol except σ. We sort the terms of the sequence in increasing order so that
ν 1 < ν 2 < L , define in as the length of (that is number of digits in) the number ν n , and
li σ n
,
as the number of σ-digits in ν n . Then the indexes {in } are nondecreasing. Denote the
corresponding sequence by ν Λ . By Lemma 3 (in the Appendix) the lim inf and lim sup of (k )
the sequence of real numbers
⎧ l ,σ ⎫ ⎧⎪ l iσ ⎫⎪ ⎪ in ⎪ , = 1 , 2 , L n ⎨ ⎬ are the same as of the sequence ⎨ ⎬ . ⎪⎩ i ⎪⎭ ⎪⎩ in ⎪⎭
The corresponding extended series
1
∑ν ν
ν∈
however is still convergent. To see this,
(k )
Λ
{
}
estimate its counting function M ( N , k ) := # ν | ν ∈ν Λ ,ν ≤ N . If r>2, there are (r-2) ways (k )
16
to select the first digit γ of the numbers in the ν Λ sequence, and ( r − 1) ways to select (k )
each subsequent ∗ -digit. By Eq. (8), and Lemmas 2 and 3, for
almost all
ν ∈ν Λ consisting of M ( M = r I with I >> 1) digits the number of ∗ -digits in ν is less (k )
than
or
equal
Mk . k +1
to
{
(k )
{
(k )
M ( N , k ) = # ν | ν ∈ν Λ ,ν ≤ N
Consequently,
< ( r − 2)
(r −1)
where ε =
−1
r−2
counting
function
} satisfies
}
I
{
M ( N , k ) =# ν | ν ∈ν Λ ,ν ≤ N = ∑ ν | ν ∈ν Λ , r I +1
the
< (r −1)
i =1
I +1
(k )
= (r − 1) (r −1)
1 , 0 < ε < 1. As the series r log r
≤ν < r
log N log r
i
}
k ⎡ ⎤ k ≤ (r − 2)∑ ⎢(r −1) +1 ⎥ i =1 ⎣ ⎦
= (r − 1) N
I
log( r −1) log r
= (r − 1) N
i
,
1−ε
∞ N 1−ε 1 = is convergent, by Theorem ∑ ∑ 2 1+ε N =1 N N =1 N ∞
1
A (of Krzyś, Powell and Šalát, [9, 10, 11])
i −1
∑ν ν
ν∈
>k>>1. Select N as N = r . We have I
17
r −1
ways.
We
obtain
different M-digit numbers if
I
{
M ( N , k ) = ∑ ν | ν ∈ν Λ , r i =1
~
r −1
I k
I
~
(k )
(log N )
r −1
i −1
}
I
≤ν < r ~ ∑k i
[k1/ log r]
log N
i r −1
i
i =1
1 and by the Gauss q = ⎢2 − log N log r ⎥⎦ ⎣
∞
M (N, k) < ∞ . By Theorem A we also have N2 N =1
∑
M (N , k) > ∞ so that N2 N =1
∑
study
−2
⎛ N +1 ⎞ ⎜ ⎟ N ⎝ ⎠
For k < r 2 , and a sufficiently large N,
have
N
2
log N − log( N +1)
⎡ r − 1 log k ⎤ ⎢2 − log N − log r ⎥ + O ⎣ ⎦
criterium
log N
2
r −1 log α log N − log( N +1) ≈ 1− ,α , ≈ 1− N log N N
(
1 N
the
]
.
⎡ ⎤ r − 1 ⎤ ⎡ log α ⎤ ⎡ 2⎤ 1 ⎡ r −1 ≈ ⎢1 − 1− 1 + ⎥ ≈ 1 + ⎢2 − − log α ⎥ + O N − 2 ⎥ ⎥ ⎢ ⎢ N ⎦⎣ N ⎦ N⎣ log N ⎣ N log N ⎦ ⎣ ⎦ = 1+
[
∑ (log N ) k 1 / log r ∞
2
⎛ log N ⎞ ⎟⎟ But ⎜⎜ ⎝ log( N + 1) ⎠
⎛ log N ⎞ ⎜⎜ ⎟⎟ ⎝ log( N + 1) ⎠
1
1
∑ ν ν
=∞.
1
∑ν
< ∞. For k > r 2 however we
ð
5. A CONJECTURE AND FINAL REMARKS
18
The use of the Stirling formula in the proof of Theorem 6 necessitated the assumption that in the limiting probability distribution of Λ, {λ0 , L, λ r −1 }, all λ j are positive (condition C). It seems likely that Theorem 6 remains valid even if some of the λi are zero. We are aware of the formal similarities between the questions discussed in this paper and the probabilistic theory of the distribution of the decimal digits of irrational numbers (Borel's normal numbers and related topics, [16-19]), of the dimensional theory of continued fractions [20], or the Hausdorff dimension of the random Cantor set [21, 22]. Discussions of this similarity however would be outside the scope of this elementary paper.
Acknowledgments. The problem was suggested by a question of Prof. Klavdia Olechko (UNAM, Mexico City) about fractals and singularities, and this paper is dedicated to her. The happy and peaceful atmosphere, and excellent research facilities, of the King Fahd
University where this research has been done, are gratefully acknowledged.
19
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Monthly 85(1978) No 8: 661-665. [5] A.D. Wadhwa, "An interesting subseries of the harmonic series", American Math.
Monthly 82(1975) No. 9: 931-933. [6] G. Korvin, Fractal Models in the Earth Sciences, Elsevier, Amsterdam, 1992. [7] J. Feder, Fractals, Plenum Press, New York, 1988. [8] G. Pólya and G. Szegö, Problems and Theorems in Analysis, Springer Verlag, BerlinHeidelberg-New York, 1978, vol. I, pp. 25-26. [9] J. Krzyś, "Oliver's theorem and its generalizations", Prace Mat., 2(1956): 159-164. [10] T. Šalát, Infinite Series, Academia, Prague, 1974, p. 101. [11] B.J. Powell and T. Šalát, "Convergence of subseries of the harmonic series and asymptotic densities of sets of integers", Publ. Inst. Math. (Beograd) 50(64)1991: 60-70. [12] I.S. Gradshtein and I.M. Ryzhik, Tables of Integrals, Sums, Series and Products. Gos. Izdatel'stvo Fiziko-Matematicheskoi Literatury, Moscow, 1963 (In Russian). [13] K. Knopp, Theory and Applications of Infinite Series, Hafner Publ. Co., New York, 1971 p. 288.
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[14] L. Barreira, B. Saussol, and J. Schmeling, "Distribution of frequencies of digits via multifractal analysis", J. Number Theory 97(2002): 413-442. [ 15 ] L. Barreira and J. Schmeling, "Sets of "non-typical" points have full topological entropy and full Hausdorff dimension", Isr. J. Math. 116(2000): 29-70. [16] E. Borel, "Sur les probabilités dénombrables et leurs applications arithmétiques",
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APPENDIX
Lemma 2. Consider the sequence of increasing integers in base r (r>2) ⎧ν 0 = γ ⎪ ⎪ν 1 = γσ ⎪ ⎪ν 2 = γσ ∗ ⎪ = γσ ∗ 2 ⎪ν 3 ⎪M ⎪ k ⎪ν k +1 = γσ ∗ (k ) ⎪ ν Λ = ⎨ν k +2 = γσ ∗k σ ⎪ ⎪M k k2 ⎪ 2 γσ σ = ∗ ν ⎪ k + k +1 ⎪M ⎪ 2 3 ⎪ν k 3 + k 2 + k +1 = γσ ∗ k σ k ∗ k ⎪ ⎪M ⎪ ⎩ where k ≥ 2 , the digit γ is different from σ and 0, the digit ∗ is not the same as σ, and we use the notation σ n = σσ σ , n ≥ 1 . Denote by l iσ the number of σ digits in L 1 42 4 3
νi
(i = 1,2,L) . Then
n − times
1 k l l lim inf i +σ1 = k + 1 ; lim sup i +σ1 = k + 1 i
i
i →∞
.
(A.1)
i →∞
Proof. To prove the first part of (A.1) we show that for any ε > 0 there exists an integer N = N (ε , k ) such that for all integers n ≥ N the following two conditions are satisfied:
(i)
l nσ 1 > −ε ; n +1 k +1
(ii) there exists an integer p, p ≥ n
for which
Select N as N = 1 + k + k 2 + ... + k 2 m +1 =
l pσ p +1
N .
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There exists a unique integer m' ≥ m for which
k
2 m ' +2
−1
k −1
1 k −ε > −ε . k +1 k +1
(A.3)
In case b), because of the construction of sequence ν Λ , the total number of digits in ν n is (k )
less than
k
2 m ' +4
−1
k −1
+ 1 , the number of σ-digits is identically
range, that is for sufficiently large values of m'
23
k
2 m ' +4
−1
k −1 2
for all n in this
−1
2 m ' +4 2 m ' +4 2 m ' +4 − 1 ⎛⎜ k − 1 ⎞⎟ −1 l nσ k − 1 ⎛⎜ k − 1 ⎞⎟ k k > ⋅ + = ⋅ + 1 1 ⎟ n +1 k 2 − 1 ⎜⎝ k − 1 k 2 − 1 k 2 m ' +4 − 1 ⎜⎝ k 2 m ' +4 ⎟⎠ ⎠ 1 ⎡ (k − 1) ⎤ 1 1 k −1 1 > − . 2 m ' +4 > −ε ⎢1 − 2 m ' +4 ⎥ = k +1 ⎢ k k +1 k +1 k k +1 ⎥ ⎣ ⎦
−1
.
(A.4)
Thus, we have proved condition (i) for both cases a) and b). As for condition (ii), in both cases a) and b) we can select the integer p as
p=k
2 m ' +4
−1
(A.5)
k −1
Then (as in Eq. A.4)
lim p →∞
2 m ' +4 − 1 ⎞⎟ − 1 ⎛⎜ k k +1 ⋅ = lim 2 ⎟ p + 1 m '→ ∞ k − 1 ⎜ k − 1 ⎠ ⎝ 2 m ' +4
l pσ
that is if m' is sufficiently large, we have indeed
To make the proof work, we should select N =
l pσ p +1
