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PureMathematics
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PublishedOnlineJanuary2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.121026
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SpectralityofSelf-SimilarMeasures
withThreeElementDigitSets
YongshenCao
SchoolofMathematicsandStatistics,FujianNormalUniversity,Fuzhou Fujian
Received:Dec.20
th
,2021;accepted:Jan.20
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,2022;published:Jan.27
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Abstract
Fu andWen provethatthe convolutionof theinfiniteBernoullimeasure generatedby
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DOI:10.12677/pm.2022.121026
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[
thecompressionratioofrealnumbers
ρ
andthesequenceofboundedthree-element
integers
D
n
=
{
0
,a
n
,b
n
}⊂
Z
isasufficientandnecessaryconditionforspectralmea-
sure.Inthispaperwestudythespectralityoftheself-similarmeasuregeneratedby
theiterativefunctionsystemdefinedbythecompressionratioofrealnumbers
ρ
and
thesetofthree-elementrealdigits
D
.Weprovethatthemeasureisspectralifand
onlyif
ρ
−
1
isanon-zerointegerwithafactorof3and
a
(
D
−
α
)
iscongruencewith
{
0
,
1
,
2
}
under(mod3)forsome
a
,where
α
∈
D.
Keywords
Self-SimilarMeasure,SpectralMeasure,Spectrum,FourierTransform
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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n
−
1
}
(
n>
1)
´
˜
k
•
¢ê
8
,
Ø
'
ρ
÷
v
|
ρ
|
=
q
p
,gcd
(
p,q
) = 1
…
2
6
q<p.
K
•
3
~
ê
a>
0
,
¦
sup
x
∈
R
{|
d
µ
ρ,D
(
x
)
|·
(ln(3+
|
x
|
))
a
}
<
+
∞
.
(3.1)
y
²
Ø
”
˜
„
5
,
·
‚
Ø
”
-
d
0
= 0
,d
1
= 1
.
Ä
K
-
C
=
{
0
,
1
,
d
2
−
d
0
d
1
−
d
0
,
···
,
d
n
−
1
−
d
0
d
1
−
d
0
}
,
K
k
|
M
D
(
x
)
|
=
|
1
n
e
2
πid
0
x
n
−
1
X
j
=0
e
2
πi
(
d
j
−
d
0
)
x
|
=
|
M
C
((
d
1
−
d
0
)
x
)
|
.
d
(2.1)
•
|
d
µ
ρ,D
((
d
1
−
d
0
)
−
1
x
)
|
=
|
d
µ
ρ,C
(
x
)
|
.
‰
½
¢ê
x
∈
R
,
K
•
3
•
˜
h
(
x
)
∈
(
−
1
2
,
1
2
]
,
¦
x
−
h
(
x
)
´
ê
.
·
‚
k
e
¡
ä
ó
.
ä
ó
:
•
3
~
ê
0
<c<
1
,
¦
X
J
÷
v
|
ρx
|
>
1
,
K
•
3
¢ê
y,
¦
|
ρx
|
>
|
y
|
>
|
ρ
|
2
·|
x
|
ln
q
ln
p
…
|
d
µ
ρ,D
(
x
)
|
<c
|
d
µ
ρ,D
(
y
)
|
.
·
‚
©
ü
«
œ
/
y
²
ä
ó
.
œ
/
˜
:
h
(
ρx
)
/
∈
(
−
1
2
p
,
1
2
p
)
.
K
|
M
D
(
ρx
)
|
=
1
n
|
1+
e
2
πiρx
+
n
−
1
X
j
=2
e
2
πid
j
ρx
|
6
n
−
2+
|
1+
e
2
πi
·
h
(
ρx
)
|
n
6
n
−
2+
|
1+
e
πi
p
|
n
.
(3.2)
-
c
=
n
−
2+
|
1+
e
πi
p
|
n
,y
=
ρx,
=
y
ä
ó
.
œ
¹
:
h
(
ρx
)
∈
(
−
1
2
p
,
1
2
p
)
.
Š
â
h
(
ρx
)
½
Â
,
ρx
−
h
(
ρx
)
k
e
Ð
m
ª
ρx
−
h
(
ρx
) =
X
j
>
0
z
j
p
j
,
Ù
¥
z
j
∈{−
1
,
0
,
1
,
···
,p
−
2
}
.
Ï
•
|
ρx
|
>
1
,
K
·
‚
Œ
-
s
>
0
•
•
ê
,
¦
z
s
6
= 0
.
d
d
Œ
h
(
ρ
s
+2
x
) =
h
ρ
s
+1
h
(
ρx
)+
z
s
q
s
+1
p
.
(3.3)
Ï
•
h
(
ρx
)
∈
−
1
2
p
,
1
2
p
…
0
<
|
ρ
|
<
1
,
·
‚
k
ρ
s
+1
h
(
ρx
)
∈
−
1
2
p
,
1
2
p
.
5
¿
gcd
(
p,q
)= 1
Ú
1
6
|
z
s
|
6
p
−
2
,
=
•
h
z
s
q
s
+1
p
>
1
p
,
l
h
(
ρ
s
+2
x
)
/
∈
−
1
2
p
,
1
2
p
.
d
(2.1)
Ú
(3.2)
•
|
d
µ
ρ,D
(
x
)
|
6
|
M
D
(
ρ
s
+2
x
)
|·|
d
µ
ρ,D
(
ρ
s
+2
x
)
|
6
c
|
d
µ
ρ,D
(
ρ
s
+2
x
)
|
.
DOI:10.12677/pm.2022.121026222
n
Ø
ê
Æ
ù
[
Ï
•
|
ρx
|
=
|
h
(
ρx
)+
X
j
>
0
z
j
p
j
|
>
p
s
−|
h
(
ρx
)
|
>
p
s
−
1
2
p
>
|
ρ
|
p
s
,
l
s
6
log
p
|
x
|
.
-
y
=
ρ
s
+2
x,
Ï
d
|
ρx
|
>
|
y
|
=
|
ρ
s
+2
x
|
>
|
ρ
|
2
|
x
|·|
ρ
|
log
p
|
x
|
=
|
ρ
|
2
|
x
|·|
x
|
log
p
|
ρ
|
=
|
ρ
|
2
|
x
|
ln
q/
ln
p
,
ä
ó
=
y
.
‰
½
x
∈
R
÷
v
|
ρx
|
>
1
,
Š
â
ä
ó
•
,
•
3
k
•
¢ê
x
1
=
x,x
2
,
···
,x
n
,
¦
|
ρx
j
|
>
|
x
j
+1
|
>
|
ρ
|
2
|
x
j
|
ln
q/
ln
p
,
|
d
µ
ρ,D
(
x
j
)
|
6
c
|
d
µ
ρ,D
(
x
j
+1
)
|
,
1
6
j
6
n
−
1
…
k
|
ρx
n
|
6
1
<
|
ρx
n
−
1
|
.
·
‚
k
|
d
µ
ρ,D
(
x
)
|
6
c
n
−
1
|
d
µ
ρ,D
(
x
n
)
|
6
c
n
−
1
max
{|
d
µ
ρ,D
(
y
)
|
:
|
ρy
|
6
1
}
(3.4)
…
1
>
|
ρx
n
|
>
|
ρ
|·|
ρ
|
2
|
x
n
−
1
|
ln
q/
ln
p
>
|
ρ
|·|
ρ
|
2+2ln
q/
ln
p
·|
x
n
−
1
|
(ln
q/
ln
p
)
2
>
···
>
|
ρ
|·|
ρ
|
2+2ln
q/
ln
p
+
···
+2(ln
q/
ln
p
)
n
−
2
·|
x
|
(ln
q/
ln
p
)
n
−
1
>
|
ρ
|·|
ρ
|
2(1
−
ln
q/
ln
p
)
−
1
·|
x
|
(ln
q/
ln
p
)
n
−
1
=
|
ρ
|·
p
−
2
·|
x
|
(ln
q/
ln
p
)
n
−
1
>
p
−
3
·|
ρx
|
(ln
q/
ln
p
)
n
−
1
.
Ï
d
3ln
p
>
(ln
q/
ln
p
)
n
−
1
·
ln
|
ρx
|
=
c
(
n
−
1)ln(ln
q/
ln
p
)
/
ln
c
·
ln
|
ρx
|
,
¤
±
[3ln
p
]
ln
c/
ln(ln
q/
ln
p
)
>
c
(
n
−
1)
·
[ln
|
ρx
|
]
ln
c/
ln(ln
q/
ln
p
)
.
a
=ln
c/
ln(ln
q/
ln
p
)
,
K
a>
0
…
[3ln
p
]
a
>
c
(
n
−
1)
·
[ln
|
ρx
|
]
a
.
Š
â
(3.4)
Œ
•
,
é
x
∈
R
,
X
J
|
ρx
|
>
1
,
K
|
d
µ
ρ,D
(
x
)
|·
[ln
|
ρx
|
]
a
6
[3ln
p
]
a
max
{|
d
µ
ρ,D
(
y
)
|
:
|
ρy
|
6
1
}
<
∞
.
q
Ï
•
sup
|
ρx
|
>
1
n
ln(3+
|
x
|
)
ln
|
ρx
|
o
<
∞
…
sup
|
ρx
|
6
1
{|
d
µ
ρ,D
(
x
)
·
[ln(3+
|
x
|
)]
a
}
<
∞
,
ù
¿
›
X
b
:= sup
x
∈
R
{|
d
µ
ρ,D
(
x
)
|·
[ln(3+
|
x
|
)]
a
}
<
∞
.
(3.5)
½
n
y
.
Ú
n
3.2(i)
D
=
{
d
0
,d
1
,d
2
}
´
n
¢ê
8
,
K
Z
(
M
D
)
6
=
∅
…
=
•
3
ê
k
1
,k
2
Ú
¢ê
DOI:10.12677/pm.2022.121026223
n
Ø
ê
Æ
ù
[
a
6
= 0
,
¦
{
d
1
−
d
0
,d
2
−
d
0
}
=
{
(3
k
1
+1)
a,
(3
k
2
+2)
a
}
…
gcd
(3
k
1
+1
,
3
k
2
+2) = 1
.
(ii)
D
=
{
0
,
3
k
1
+1
,
3
k
2
+2
}
,
Ù
¥
k
1
,k
2
∈
Z
.
X
J
gcd
(3
k
1
+1
,
3
k
2
+2)=1
,
K
Z
(
M
D
)=
±
1
3
+
Z
=
1
3
(
Z
\
3
Z
)
.
y
²
(i)
Š
â
M
D
(
x
)
½
Â
Œ
±
w
Ñ
M
D
(
x
) =
1
3
(
e
−
2
πid
0
x
+
e
−
2
πid
1
x
+
e
−
2
πid
2
x
) = 0
,
⇔
cos
(2
π
(
d
1
−
d
0
)
x
)+
cos
(2
π
(
d
2
−
d
0
)
x
) =
−
1
,
sin
(2
π
(
d
1
−
d
0
)
x
) =
−
sin
(2
π
(
d
2
−
d
0
)
x
)
,
⇔
•
3
ê
n
1
,n
2
,
¦
(
d
1
−
d
0
)
x
=
n
1
±
1
3
,
(
d
1
+
d
2
−
2
d
0
)
x
=
n
1
+
n
2
,
⇔
•
3
ê
n
1
,n
2
,
¦
{
3(
d
1
−
d
0
)
x,
3(
d
2
−
d
0
)
x
}
=
{
3
n
1
+1
,
3
n
2
+2
}
.
(3.6)
w
,
•
3
ê
k
1
,k
2
,
¦
1
gcd
(3
n
1
+1
,
3
n
2
+2)
{
3
n
1
+1
,
3
n
2
+2
}
=
{
3
k
1
+1
,
3
k
2
+2
}
.
ù
V
«
gcd
(3
k
1
+1
,
3
k
2
+2) = 1
.
(i)
=
y
.
(ii)
-
x
∈
Z
(
M
D
)
,
K
(3.6)
V
«
•
3
ê
n
1
,n
2
Ú
l
∈{−
1
,
1
}
,
¦
(3
k
1
+ 1)
x
=
n
1
+
l
3
,
(3
k
2
+2)
x
=
n
2
−
l
3
.
Ï
d
x
−
l
3
=
n
1
−
lk
1
3
k
1
+1
=
n
2
−
lk
2
3
k
2
+2
.
Š
â
gcd
(3
k
1
+1
,
3
k
2
+2) = 1
Œ
•
(3
k
1
+1)
|
(
n
1
−
lk
1
)
…
(3
k
2
+2)
|
(
n
2
−
lk
2
)
.
ù
¿
›
X
x
∈±
1
3
+
Z
.
Ï
d
Z
(
M
D
)
⊆±
1
3
+
Z
.
x
=
l
3
+
z
∈±
1
3
+
Z
,
Ù
¥
z
∈
Z
,l
∈{−
1
,
1
}
.
K
(3
k
1
+ 1)
x
=(3
k
1
+ 1)
z
+
lk
1
+
l
3
…
[(3
k
1
+1)+(3
k
2
+2)]
x
= (
k
1
+
k
2
+1)(
l
+3
z
)
.
K
Š
â
(3.6)
Œ
•
M
D
(
x
) = 0
.
Ï
d
Z
(
M
D
)
⊇±
1
3
+
Z
,
l
Z
(
M
D
) =
±
1
3
+
Z
.
Š
â
Ú
n
3
.
2
Œ
•
Z
(
M
D
)
Ø
¹
k
1
,k
2
,
l
k
e
ã
í
Ø
.
í
Ø
3.3
D
=
{
0
,
3
k
1
+1
,
3
k
2
+2
}
…
gcd
(3
k
1
+1
,
3
k
2
+2)=1
,
Ù
¥
k
1
,k
2
∈
Z
.
X
J
C
=
{
0
,
1
,
2
}
,
K
d
µ
ρ,C
(
x
) = 0
…
=
d
µ
ρ,D
(
x
) = 0
.
Ú
n
3.4
D
=
{
0
,
3
k
1
+1
,
3
k
2
+2
}
…
gcd
(3
k
1
+1
,
3
k
2
+2) = 1
,
Ù
¥
k
1
,k
2
∈
Z
.
X
J
µ
ρ,D
´
Ì
ÿ
Ý
,
K
•
3
ê
p,q,
¦
|
ρ
|
=
q
3
p
…
gcd
(
q,
3
p
) = 1
.
y
²
Λ
´
µ
ρ,D
Ì
,
Š
â
í
Ø
3
.
3
Œ
•
E
Λ
´
L
2
(
µ
ρ,C
)
Ã
¡
8
,
Ù
¥
C
=
{
0
,
1
,
2
}
.
2
¦
^
[4]
¥
½
n
1
.
2
Œ
•
•
3
ê
p,q,r,
¦
|
ρ
|
=(
q
3
p
)
1
/r
…
gcd
(
q,
3
p
)= 1
.
Ø
”
˜
„
5
,
b
r
´¦
(
q
3
p
)
1
/r
∈
Q
•
ê
,
=
?
¿
ê
k<r,
(
q
3
p
)
1
/k
´
Ã
n
ê
,
K
|
ρ
|
•
õ
‘
ª´
3
px
r
−
q.
Ï
•
µ
=
δ
ρD
∗
[
δ
ρ
2
D
∗
δ
ρ
3
D
∗···∗
δ
ρ
n
D
∗···
]
,µ
=
δ
ρ
2
D
∗
[
δ
ρD
∗
δ
ρ
3
D
∗···∗
δ
ρ
n
D
∗···
]
,
Š
â
Ú
n
2
.
4
Œ
•
(Λ
−
Λ)
∩
Z
(
M
ρD
)
6
=
∅
,
(Λ
−
Λ)
∩
Z
(
M
ρ
2
D
)
6
=
∅
.
DOI:10.12677/pm.2022.121026224
n
Ø
ê
Æ
ù
[
Ï
d
,
Ï
L
Ú
n
3
.
2
Œ
•
•
3
λ
j
∈
Λ(
j
= 0
,
1
,
2
,
3)
Ú
z
1
,z
2
∈
Z
,
¦
λ
0
−
λ
1
=
ρ
−
1
(
±
1
3
+
z
1
)
,λ
2
−
λ
3
=
ρ
−
2
(
±
1
3
+
z
2
)
.
(3.7)
,
˜
•
¡
,
é
λ
∈
Λ
\{
λ
0
,λ
1
}
,
k
λ
−
λ
0
,λ
−
λ
1
∈
Z
(
d
µ
ρ,D
) =
S
∞
n
=1
ρ
−
n
Z
(
M
D
)
.
¤
±
,
•
3
ê
n
0
>
0
,n
1
>
0
,z
3
,z
4
,
¦
λ
−
λ
0
=
ρ
−
n
0
(
±
1
3
+
z
3
)
,λ
−
λ
1
=
ρ
−
n
1
(
±
1
3
+
z
4
)
.
Ï
d
ρ
−
n
1
(
±
1
3
+
z
4
)
−
ρ
−
n
0
(
±
1
3
+
z
3
) =
λ
0
−
λ
1
=
ρ
−
1
(
±
1
3
+
z
1
)
.
Ø
”
˜
„
5
,
b
n
1
>
n
0
>
1
,
K
ρ
´
•
§
(
±
1+3
z
1
)
x
n
1
−
1
+(
±
1+3
z
3
)
x
n
1
−
n
0
−
(
±
1+3
z
4
) = 0(3.8)
)
.
n
1
−
1=
l
1
r
+
s
1
,n
1
−
n
0
=
l
2
r
+
s
2
,
Ù
¥
l
1
>
0
,l
2
>
0
,
0
6
s
1
,s
2
<r.
K
(3.8)
Ú
ρ
=
±
(
q
3
p
)
1
/r
V
«
•
3
ê
m
1
,m
2
,m
3
,
Ù
¥
m
3
6
= 0
,
¦
ρ
´
ª
m
1
x
s
1
+
m
2
x
s
2
−
m
3
= 0
)
.
Ï
•
|
ρ
|
•
õ
‘
ª´
3
px
r
−
q
…
0
6
s
1
,s
2
<r,
Œ
±
s
1
=
s
2
=0
.
l
r
|
(
n
0
−
1)
,r
|
(
n
1
−
1)
…
(Λ
\{
λ
0
,λ
1
}
)
−
λ
0
⊂{
0
}∪
∞
[
n
=0
ρ
−
(1+
nr
)
Z
(
M
D
)
!
,
2
Š
â
(3.7)
1
˜
‡
ª
f
,
k
Λ
−
λ
0
⊂{
0
}∪
∞
[
n
=0
ρ
−
(1+
nr
)
Z
(
M
D
)
!
.
(3.9)
é
α
6
=
β
∈
Λ
\{
λ
0
}
,
Ï
L
(3.9)
Œ
•
•
3
ê
n
2
,n
3
,z
5
,z
6
¦
α
−
λ
0
=
ρ
−
(1+
n
2
r
)
(
±
1
3
+
z
5
)
,β
−
λ
0
=
ρ
−
(1+
n
3
r
)
(
±
1
3
+
z
6
)
.
q
Ï
•
α
−
β
∈
Z
(
d
µ
ρ,D
) =
S
∞
n
=1
ρ
−
n
Z
(
M
D
)
,
l
•
3
ê
n,z
7
,
¦
ρ
−
(1+
n
2
r
)
(
±
1
3
+
z
5
)
−
ρ
−
(1+
n
3
r
)
(
±
1
3
+
z
6
) =
ρ
−
n
(
±
1
3
+
z
7
)
.
Œ
±
a
q
r
|
(
n
−
1)
.
Ï
d
,
α
−
β
∈
S
∞
n
=0
ρ
−
(1+
nr
)
Z
(
M
D
)
,
¤
±
Λ
−
Λ
⊂{
0
}∪
∞
[
n
=0
ρ
−
(1+
nr
)
Z
(
M
D
)
!
.
Ï
L
(3.7)
1
‡
ª
f
Œ
•
r
= 1
.
Ï
d
,
•
3
ê
p,q,
¦
|
ρ
|
=
q
3
p
…
gcd
(
q,
3
p
) = 1
.
X
Ã
A
Ï
`
²
,
e
©
¥
-
D
=
{
0
,
3
k
1
+1
,
3
k
2
+2
}
,
|
ρ
|
=
q
3
p
,
Ù
¥
k
1
,k
2
∈
Z
,gcd
(3
k
1
+1
,
3
k
2
+
DOI:10.12677/pm.2022.121026225
n
Ø
ê
Æ
ù
[
2) = 1
,p,q
´
ê
…
gcd
(
q,
3
p
)= 1
.
Ú
n
3.5
X
J
Λ
´
µ
ρ,D
Ì
,
Ù
¥
0
∈
Λ
.
K
(Λ
−
Λ)
\{
0
}⊂
∞
[
n
=0
(3
p
)
n
(
Z
\
3
Z
)
3
|
ρ
|
.
(3.10)
y
²
Ï
•
µ
=
δ
ρD
∗
[
δ
ρ
2
D
∗
δ
ρ
3
D
∗···∗
δ
ρ
n
D
∗···
]
,
l
d
Ú
n
2
.
4
Œ
•
(Λ
−
Λ)
∩
Z
(
M
ρD
)
6
=
∅
.
Ï
d
,
Ï
L
Ú
n
3
.
2
Œ
•
•
3
λ
0
,λ
1
∈
Λ
Ú
k
0
∈
(
Z
\
3
Z
)
,
¦
λ
0
−
λ
1
=
k
0
3
|
ρ
|
.
Ä
k
·
‚
y
²
Λ
−
λ
1
⊂{
0
}∪
S
∞
n
=0
(3
p
)
n
(
Z
\
3
Z
)
3
|
ρ
|
.
é
λ
∈
Λ
\{
λ
0
,λ
1
}
,
Š
â
(2.2)
,
(2.3)
Ú
Ú
n
2
.
4
Œ
•
•
3
ê
k
1
,k
2
∈
(
Z
\
3
Z
)
Ú
n
1
,n
2
>
0
,
¦
λ
−
λ
1
=
k
1
3
|
ρ
|
n
1
,λ
−
λ
0
=
k
2
3
|
ρ
|
n
2
.
l
k
1
3
|
ρ
|
n
1
−
k
0
3
|
ρ
|
=
λ
−
λ
0
=
k
2
3
|
ρ
|
n
2
.
q
Ï
•
|
ρ
|
=
q
3
p
,
l
þ
¡
ª
f
d
u
k
1
(3
p
)
n
1
−
1
q
n
1
−
1
−
k
0
=
k
2
(3
p
)
n
2
−
1
q
n
2
−
1
.
·
‚
©
n
«
œ
/
?
Ø
:
œ
¹
˜
:
n
1
>
1
…
n
2
>
1
.
Ï
•
gcd
(
q,
3
p
) = 1
,
¤
±
þ
¡
Ø
ª
V
«
3
p
|
k
0
,
ù
†
k
0
∈
(
Z
\
3
Z
)
g
ñ
.
œ
¹
:
n
1
>
1
…
n
2
= 1
.
K
k
1
(3
p
)
n
1
−
1
q
n
1
−
1
−
k
0
=
k
2
,
l
k
1
q
n
1
−
1
´
ê
.
ù
¿
›
X
•
3
ê
n>
0
,
¦
λ
−
λ
1
∈
(3
p
)
n
(
Z
\
3
Z
)
3
|
ρ
|
.
œ
¹
n
:
n
1
= 1
.
w
,
λ
−
λ
1
∈
(
Z
\
3
Z
)
3
|
ρ
|
.
n
þ
=
Λ
−
λ
1
⊂{
0
}∪
∞
[
n
=0
(3
p
)
n
(
Z
\
3
Z
)
3
|
ρ
|
!
.
(3.11)
,
·
‚
y
²
(3.10)
.
é
Λ
\{
λ
1
}
¥
ü
‡
Ø
Ó
ƒ
λ
=
λ
1
+
(3
p
)
m
l
1
3
|
ρ
|
,λ
0
=
λ
1
+
(3
p
)
k
l
2
3
|
ρ
|
,
Ù
¥
l
1
,l
2
∈
(
Z
\
3
Z
)
,m,k
∈
Z
.
Ï
L
(2.3)
Œ
•
•
3
ê
s
>
0
Ú
l
3
∈
(
Z
\
3
Z
)
,
¦
(3
p
)
m
l
1
3
|
ρ
|
−
(3
p
)
k
l
2
3
|
ρ
|
=
(3
p
)
s
l
3
3
|
ρ
|
q
s
.
DOI:10.12677/pm.2022.121026226
n
Ø
ê
Æ
ù
[
Ï
d
q
s
|
l
3
,
ù
¿
›
X
λ
−
λ
0
á
u
(3.11)
m
ý
,
l
(Λ
−
Λ)
⊂{
0
}∪
∞
[
n
=0
(3
p
)
n
(
Z
\
3
Z
)
3
|
ρ
|
!
.
Ú
n
3.6
Λ
´
µ
ρ,D
Ì
,
Ù
¥
0
∈
Λ
.
K
•
3
ê
z
j
Ú
Λ
j
⊂
Z
(
j
=0
,
1
,
2)
…
0
∈
Λ
j
,
¦
Λ
j
´
µ
ρ,D
Ì
…
Λ
k
X
e
©
)
Λ =
2
[
j
=0
[
j
+3
z
j
3
ρ
+
ρ
−
1
Λ
j
]
.
(3.12)
y
²
Š
â
Ú
n
3
.
5
Œ
•
Λ
⊂
Z
3
ρ
.
é
j
∈{
0
,
1
,
2
}
,
X
J
Λ
∩
j
+3
Z
3
ρ
6
=
∅
,
K
•
3
λ
j
=
j
+3
z
j
3
ρ
∈
Λ
,
¦
|
j
+3
z
j
3
ρ
|
= min
{|
λ
|
:
λ
∈
Λ
∩
j
+3
Z
3
ρ
}
,j
= 0
,
1
,
2
.
-
Λ
j
=
ρ
(Λ
−
λ
j
)
∩
Z
,j
= 0
,
1
,
2
.
(3.13)
w
,
0
∈
Λ
j
…
Λ =
2
[
j
=0
[
λ
j
+
ρ
−
1
Λ
j
](3.14)
´
Ø
¿
,
=
i
6
=
j
∈{
0
,
1
,
2
}
ž
,
k
[
λ
j
+
ρ
−
1
Λ
j
]
∩
[
λ
i
+
ρ
−
1
Λ
i
] =
∅
.
Š
â
Ú
n
2
.
1
Ú
(3.14)
Œ
•
,
é
?
¿
x
∈
R
,
·
‚
k
1 =
X
λ
∈
Λ
|
d
µ
ρ,D
(
λ
+
x
)
|
2
=
2
X
j
=0
X
γ
j
∈
Λ
j
|
d
µ
ρ,D
(
λ
j
+
ρ
−
1
γ
j
+
x
)
|
2
=
2
X
j
=0
|
M
D
(
j
3
+
ρx
)
|
2
X
γ
j
∈
Λ
j
|
d
µ
ρ,D
(
j
+3
z
j
3
+
γ
j
+
ρx
)
|
2
.
(3.15)
‰
½
j
∈{
0
,
1
,
2
}
,
e
Λ
j
6
=
∅
,
?
γ
j
6
=
γ
0
j
∈
Λ
j
,
(3.13)
L
²
λ
j
+
ρ
−
1
γ
j
6
=
λ
j
+
ρ
−
1
γ
0
j
∈
Λ
.
Ï
d
(
λ
j
+
ρ
−
1
γ
j
)
−
(
λ
j
+
ρ
−
1
γ
0
j
)
∈
Z
(
d
µ
ρ,D
)
,
l
•
3
ê
z
∈
(
Z
\
3
Z
)
Ú
n>
0
,
¦
(
λ
j
+
ρ
−
1
γ
j
)
−
(
λ
j
+
ρ
−
1
γ
0
j
) =
z
3
|
ρ
|
n
.
Ï
d
γ
j
−
γ
0
j
=
z
3
|
ρ
|
n
−
1
,
q
Ï
•
γ
j
−
γ
0
j
´
ê
,
l
n>
1
.
ù
¿
›
X
γ
j
−
γ
0
j
∈
Z
(
d
µ
ρ,D
)
,
l
E
Λ
j
´
˜
8
½
ö
´
L
2
(
µ
ρ,D
)
´
8
.
q
Ï
•
(
D,
{
0
,
1
3
,
2
3
}
)
´
ƒ
N
é
,
Š
â
Ú
n
2
.
1
Ú
(3.15)
Œ
•
1 =
2
X
j
=0
|
M
D
(
j
3
+
ρx
)
|
2
X
γ
j
∈
Λ
j
|
d
µ
ρ,D
(
j
+3
z
j
3
+
γ
j
+
ρx
)
|
2
6
2
X
j
=0
|
M
D
(
j
3
+
ρx
)
|
2
= 1
.
q
Ï
•
A
¤
k
x
∈
R
,
k
M
D
(
j
3
+
ρx
)
6
= 0
,
l
X
γ
j
∈
Λ
j
|
d
µ
ρ,D
(
j
+3
z
j
3
+
γ
j
+
ρx
)
|
2
≡
1
,
∀
x
∈
R
,j
= 0
,
1
,
2
.
DOI:10.12677/pm.2022.121026227
n
Ø
ê
Æ
ù
[
Š
â
Ú
n
2
.
1
Œ
•
Λ
j
´
µ
ρ,D
(
j
= 0
,
1
,
2)
Ì
,
Ú
n
y
.
y
²
½
n
1
.
1
¿
©
5
:
d
^
‡
•
|
ρ
|
−
1
∈
3
N
+
…
•
3
ê
k
1
,k
2
Ú
¢ê
a
6
=0
,
¦
{
d
1
−
d
0
,d
2
−
d
0
}
=
{
(3
k
1
+1)
a,
(3
k
2
+2)
a
}
.
C
=
{
0
,
3
k
1
+1
,
3
k
2
+2
}
,
K
(
C,
{−
1
3
,
0
,
1
3
}
)
´
ƒ
N
é
.
Ï
d
(3
p,C,
{−
p,
0
,p
}
)
´
Hadamard
n
é
,
Š
â
[5]
Œ
•
µ
ρ,C
´
Ì
ÿ
Ý
.
•
d
·
K
2
.
5
Œ
•
µ
ρ,D
´
Ì
ÿ
Ý
.
7
‡
5
:
X
J
µ
ρ,D
´
Ì
ÿ
Ý
,
K
•
3
Œ
ê
8
Λ
⊂
R
,
¦
(
µ
ρ,D
,
Λ)
´
Ì
é
,
Ø
”
˜
„
5
,
·
‚
b
0
∈
Λ
.
K
Λ
\{
0
}⊂
Z
(
d
µ
ρ,D
)
,
l
Z
(
M
D
)
6
=
∅
.
d
Ú
n
3
.
2
y
²
•
3
ê
k
1
,k
2
Ú
¢ê
a
6
=0
,
¦
{
d
1
−
d
0
,d
2
−
d
0
}
=
{
(3
k
1
+1)
a,
(3
k
2
+2)
a
}
…
gcd
(3
k
1
+1
,
3
k
2
+2) = 1
.
Ï
d
·
K
2
.
5
y
²
µ
ρ,C
´
Ì
ÿ
Ý
,
Ù
¥
C
=
{
0
,
3
k
1
+1
,
3
k
2
+2
}
.
ù
¿
›
X
•
I
‡
•
Ä
D
=
{
0
,
3
k
1
+1
,
3
k
2
+2
}
œ
¹
e
|
ρ
|
−
1
∈
3
N
+
¤
á
=
Œ
.
D
=
{
0
,
3
k
1
+1
,
3
k
2
+2
}
ž
,
Š
â
Ú
n
3
.
4
Œ
•
•
3
ê
p,q,
¦
|
ρ
|
=
q
3
p
…
gcd
(
q,
3
p
) = 1
.
Ú
n
3
.
6
y
²
•
3
Œ
ê
8
0
∈
Λ
⊂
Z
,
¦
(
µ
ρ,D
,
Λ)
´
Ì
é
.
•
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DOI:10.12677/pm.2022.121026230
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