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PureMathematicsnØêÆ,2022,12(1),218-232
PublishedOnlineJanuary2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.121026
nêi8gƒqÿÝÌ55Ÿ
ùùù[[[
4ŒÆêƆÚOÆ§4ï4²
ÂvFϵ2021c1220F¶¹^Fϵ2022c120F¶uÙFϵ2022c127F
Á‡
FuÚWeny²Ø '•¢êρÚk.nêi8D
n
={0,a
n
,b
n
}⊂Z)¤á
BernoulliòÈÿÝ´ÌÿÝ¿‡^‡. ©ïÄdØ '•¢êρÚn¢êi8D½ÂS
“¼êXÚ)¤gƒqÿÝÌ5Ÿ, ·‚y²TÿÝ´ÌÿÝ…=ρ
−1
´±3•Ïfš
"ê…•3š"¢êa,¦a(D−α) 3Ó{8Ü{0,1,2 },Ù¥α∈D"
'…c
gƒqÿݧÌÿݧ̧FourierC†
SpectralityofSelf-SimilarMeasures
withThreeElementDigitSets
YongshenCao
SchoolofMathematicsandStatistics,FujianNormalUniversity,Fuzhou Fujian
Received:Dec.20
th
,2021;accepted:Jan.20
th
,2022;published:Jan.27
th
,2022
Abstract
Fu andWen provethatthe convolutionof theinfiniteBernoullimeasure generatedby
©ÙÚ^:ù[.nêi8gƒqÿÝÌ55Ÿ[J].nØêÆ,2022,12(1):218-232.
DOI:10.12677/pm.2022.121026
ù[
thecompressionratioofrealnumbersρandthesequenceofboundedthree-element
integersD
n
={0,a
n
,b
n
}⊂Zisasufficientandnecessaryconditionforspectralmea-
sure.Inthispaperwestudythespectralityoftheself-similarmeasuregeneratedby
theiterativefunctionsystemdefinedbythecompressionratioofrealnumbersρand
thesetofthree-elementrealdigitsD.Weprovethatthemeasureisspectralifand
onlyifρ
−1
isanon-zerointegerwithafactorof3anda(D−α)iscongruencewith
{0,1,2 }under(mod3)forsomea,whereα∈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/
1.Úó9̇(J
¡R
d
þ˜‡BorelVÇÿݵ´˜‡ÌÿÝ,XJ•3Λ⊂R
d
¦•ê¼ê8E
Λ
:=
{e
2πihλ,xi
: λ∈Λ}´L
2
(µ) ˜‡Ä, dž¡Λ´µ˜‡Ì. Ω ´R
d
¥äkLebesgue
ÿÝf8, XJLebesgueÿÝ•›3Ω þ´˜‡ÌÿÝ, K¡Ω ´˜‡Ì8. 3[7] ¥FugledeJ
ÑͶÌ8ߎµ
Ì8ߎ8ÜΩ´˜‡Ì8…=Ω´˜‡²£tile,=•3Γ⊂R
d
¦ª
P
t∈Γ
I
Ω
(x−t) = 1'uLebesgueÿÝA??¤á.
3p‘œ/(d>3) ¥,®k(J[10] [11][14]L²Tߎؤá. d= 1½2 ž,TߎE,
´m˜¯K.5¿8ÜΩ´Ä•Ì8†ÿÝÌ5k',<‚g,¬¯ŸoÿÝ´ÌÿÝ?
1998c,JorgensenÚPedersen3[9] ¥uy1˜‡ÛÉ!šfÌÿÝ. ¦‚y²Ø '•
1
2k
áBernoulliòÈÿÝ´ÌÿÝ, ¿EуAÌ, Ø '•
1
3
áBernoulliòÈ
ÿÝØ´ÌÿÝ. ù˜¯<uy ¦<‚kŒUò²;Fourier©Ûïá3©/8þ, •mM
g•ÿÝÚMoran.g•ÿÝÌ5ïÄ#+•.'uù•¡ïÄ®k´L(J,X[1–15].
0 <|ρ|<1,D= {d
1
,d
2
,d
3
}⊂R. K
µ
ρ,D
:= δ
ρD
∗δ
ρ
2
D
∗···∗δ
ρ
n
D
∗···(1.1)
½Â˜‡RþVÇÿݧ·‚¡ƒ•gƒqÿÝ"©•Äµ
ρ,D
Ì5Ÿ§y²Xe(
DOI:10.12677/pm.2022.121026219nØêÆ
ù[
Jµ
½n1.10 <|ρ|<1,D= {d
0
,d
1
,d
2
}⊂R´n¢ê8,µ
ρ,D
d(1.1)½Â.Kµ
ρ,D
´ÌÿÝ
…=|ρ|
−1
∈3N
+
…•3êk
1
,k
2
Ú¢êa6= 0,¦{d
1
−d
0
,d
2
−d
0
}= {(3k
1
+1)a,(3k
2
+2)a}.
éuù‡¯K, ®²k¤J´: (1) D={0,1,2}ž, ©z[2], [4]y²: µ
ρ,D
´ÌÿÝ
…=|ρ|
−1
∈3N
+
.(2)D={0,a,b}⊂Zž, ©z[6], [13]y²:µ
ρ,D
´ÌÿÝ…=
|ρ|
−1
∈3N
+
…•3êk
1
,k
2
Úêγ6= 0¦{a,b}= {(3k
1
+1)γ,(3k
2
+2)γ}(ù´©z[6],
[13](JAÏœ¹).
XJD= {d
0
,d
1
,d
2
}⊂R´?¿¢ê8( nêi8), Kµ
ρ,D
Ì5Ÿ„´˜‡ ™)û
¯K. ·‚þã½n)ûù‡¯K. I‡5¿,3[6] ¥,FuÚWen•Ä{D
n
}
∞
n=1
´•¹"
nêS,Šânê8Mask¼ê":8ÚØÓœ¹eρ•õ‘ª•(Ø.
·‚ùŸØ©•ÄD•n¢ê8, Äky²Moran.ÿÝ'uêi 8 ²£Ú ˜Ì5ØC5,
,òÿÝÌ©)•n‡8Ü,•¦^‡y{(Ø.
2.ÄÚn
3ù˜!,·‚̇0Ï^ÎÒÚ˜®•(J.
D´k•¢ê8,¡
M
D
(x) =
1
#D
X
d∈D
e
−2πidx
,x∈R
•DMask¼ê.µ
ρ,D
´(1.1)½ÂBorelVÇÿÝ,Kµ
ρ,D
FourierC†•:
dµ
ρ,D
(ξ) =
Z
e
−2πiξx
dµ
ρ,D
(x) =
∞
Y
n=1
M
D
(ρ
n
ξ),ξ∈R.(2.1)
PZ(f)•¼êf":8, K
Z(dµ
ρ,D
) =
∞
[
n=1

ρ
−n
Z(M
D
)

.(2.2)
Λ´˜¢êf8,PE
Λ
:= {e
2πiλx
: λ∈Λ}. K•ê¼ê8E
Λ
´L
2
(µ
ρ,D
)8…=
Λ−Λ ⊂{0}∪Z(dµ
ρ,D
).(2.3)
ePQ
Λ
(ξ) :=
P
λ∈Λ
|dµ
ρ,D
(λ+ξ)|
2
,|^Parsevalª,Œe¡äE
Λ
5-‡óä.
Ún2.1[9]•ê¼ê8E
Λ
´L
2
(µ
ρ,D
)8…=é?¿ξ∈R,kQ
Λ
(ξ)61.E
Λ
´
L
2
(µ
ρ,D
)Ä…=Q
Λ
(ξ) ≡1.
½Â2.2D,C´ü‡Äêƒk•¢ê8,XJ
H:=
1
√
#D

e
2πidc

d∈D,c∈C
DOI:10.12677/pm.2022.121026220nØêÆ
ù[
´jÝ,K¡(D,C) ´˜‡ƒNé.
ŠâÚn2.1,Œe¡ƒNéäOK.
Ún2.3D,C´k•¢ê8,K(D,C) ´ƒNé…=#D= #C…
X
c∈C
|M
D
(ξ+c)|
2
≡1,ξ∈R.
Ún2.4µ
1
,µ
2
´BorelVÇÿÝ, Ù¥|cµ
1
(ξ)|=1 )8´RlÑf8. XJΛ ´µ
1
∗µ
2
Ì,K•3α,β∈Λ,¦cµ
1
(α−β) = 0.
y²Ï•Λ ´µ
1
∗µ
2
Ì,l
\
µ
1
∗µ
2
(α−β) = 0,α6= β∈Λ.(2.4)
2ŠâÿÝòÈÚÿÝFourierC†½Â,k
\
µ
1
∗µ
2
(x) = cµ
1
(x)cµ
2
(x).(2.5)
b?¿α,β∈Λ,kcµ
1
(α−β)6= 0,KŠâ(2.4)Ú(2.5) Œ•cµ
2
(α−β)= 0,lE
Λ
´L
2
(µ
2
)
8.Ïd,ŠâÚn2.1Œ•
X
λ∈Λ
|cµ
2
(λ+ξ)|
2
61,∀ξ∈R.
Ï•|cµ
1
(ξ)|= 1)8´RlÑf8…Λ´Œê8,l•3ξ∈R,¦
|cµ
1
(λ+ξ)|
2
<1,∀λ∈Λ.
2d(2.5)Υ
1 = Q
Λ
(x) =
X
λ∈Λ
|cµ
1
(λ+ξ)|
2
|cµ
2
(λ+ξ)|
2
<
X
λ∈Λ
|cµ
2
(λ+ξ)|
2
61,
ù´gñ,lÚny.
3!•,·‚0Moran.ÿÝ'uêi8²£Ú ˜Ì5ØC5.
·K2.50 <|ρ|<1,D= {d
0
,d
1
,···,d
n−1
}⊂R(n>1),µ
ρ,D
´d(1.1)½ÂgƒqÿÝ.
XJC= {0,a(d
1
−d
0
),···,a(d
n−1
−d
0
)},Ù¥a6= 0.Kµ
ρ,D
´ÌÿÝ…=µ
ρ,C
´ÌÿÝ.
y²é?¿¢ê8Λ,Γ = a
−1
Λ.Ï•é?¿Borel8Ak
µ
ρ,D
(A) = µ
ρ,C
(a(A−
∞
X
n=1
ρ
n
d
0
)),
l|dµ
ρ,D
(x)|= |dµ
ρ,C
(a
−1
x)|.Ïd
X
λ∈Λ
|dµ
ρ,D
(x+λ)|
2
=
X
λ∈Λ
|dµ
ρ,C
(a
−1
x+a
−1
λ)|
2
=
X
γ∈Γ
|dµ
ρ,C
(a
−1
x+γ)|
2
.
DOI:10.12677/pm.2022.121026221nØêÆ
ù[
ùV«3Rþ
P
λ∈Λ
|dµ
ρ,D
(x+λ)|
2
≡1…=3Rþ
P
γ∈Γ
|dµ
ρ,C
(y+γ)|
2
≡1.ŠâÚn2.1½
ny.
3.̇½ny²
Šâ[6]¥(ØÚ·K2.5,·‚®²½n1.1y².©‰Ñ,˜«y²•{.
½n3.1µ
ρ,D
d(1.1)½Â,Ù¥D= {d
0
,d
1
,···,d
n−1
}(n>1) ´˜k•¢ê8,Ø 'ρ÷
v|ρ|=
q
p
,gcd(p,q) = 1 …2 6q<p.K•3~êa>0,¦
sup
x∈R
{|dµ
ρ,D
(x)|·(ln(3+|x|))
a
}<+∞.(3.1)
y²Ø”˜„5, ·‚Ø”-d
0
= 0,d
1
= 1.ÄK-C= {0,1,
d
2
−d
0
d
1
−d
0
,···,
d
n−1
−d
0
d
1
−d
0
},Kk
|M
D
(x)|= |
1
n
e
2πid
0
x
n−1
X
j=0
e
2πi(d
j
−d
0
)x
|= |M
C
((d
1
−d
0
)x)|.
d(2.1)•|dµ
ρ,D
((d
1
−d
0
)
−1
x)|= |dµ
ρ,C
(x)|.
‰½¢êx∈R,K•3•˜h(x) ∈(−
1
2
,
1
2
],¦x−h(x)´ê.·‚ke¡äó.
äó: •3~ê0<c<1,¦XJ÷v|ρx|>1,K•3¢êy,¦|ρx|>|y|>|ρ|
2
·|x|
lnq
lnp
…
|dµ
ρ,D
(x)|<c|dµ
ρ,D
(y)|.
·‚©ü«œ/y²äó.
œ/˜:h(ρx)/∈(−
1
2p
,
1
2p
).K
|M
D
(ρx)|=
1
n
|1+e
2πiρx
+
n−1
X
j=2
e
2πid
j
ρx
|6
n−2+|1+e
2πi·h(ρx)
|
n
6
n−2+|1+e
πi
p
|
n
.(3.2)
-c=
n−2+|1+e
πi
p
|
n
,y= ρx,=yäó.
œ¹:h(ρx) ∈(−
1
2p
,
1
2p
).Šâh(ρx)½Â,ρx−h(ρx)keÐmª
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.dd
Œ
h(ρ
s+2
x) = h

ρ
s+1
h(ρx)+
z
s
q
s+1
p

.(3.3)
Ï•h(ρx) ∈

−
1
2p
,
1
2p

…0 <|ρ|<1,·‚kρ
s+1
h(ρx) ∈

−
1
2p
,
1
2p

.5¿gcd(p,q)= 1 Ú
1 6|z
s
|6p−2,=•



h

z
s
q
s+1
p




>
1
p
,lh(ρ
s+2
x)/∈

−
1
2p
,
1
2p

.d(2.1)Ú(3.2)•
|dµ
ρ,D
(x)|6|M
D
(ρ
s+2
x)|·|dµ
ρ,D
(ρ
s+2
x)|6c|dµ
ρ,D
(ρ
s+2
x)|.
DOI:10.12677/pm.2022.121026222nØêÆ
ù[
Ï•
|ρx|= |h(ρx)+
X
j>0
z
j
p
j
|>p
s
−|h(ρx)|>p
s
−
1
2p
>|ρ|p
s
,
ls6log
p
|x|.-y= ρ
s+2
x,Ïd
|ρx|>|y|= |ρ
s+2
x|>|ρ|
2
|x|·|ρ|
log
p
|x|
= |ρ|
2
|x|·|x|
log
p
|ρ|
= |ρ|
2
|x|
lnq/lnp
,
äó=y.
‰½x∈R÷v|ρx|>1,Šâäó•,•3k•¢êx
1
= x,x
2
,···,x
n
,¦
|ρx
j
|>|x
j+1
|>|ρ|
2
|x
j
|
lnq/lnp
,|dµ
ρ,D
(x
j
)|6c|dµ
ρ,D
(x
j+1
)|,1 6j6n−1
…k
|ρx
n
|61 <|ρx
n−1
|.
·‚k
|dµ
ρ,D
(x)|6c
n−1
|dµ
ρ,D
(x
n
)|6c
n−1
max{|dµ
ρ,D
(y)|: |ρy|61}(3.4)
…
1 >|ρx
n
|>|ρ|·|ρ|
2
|x
n−1
|
lnq/lnp
>|ρ|·|ρ|
2+2lnq/lnp
·|x
n−1
|
(lnq/lnp)
2
>···>|ρ|·|ρ|
2+2lnq/lnp+···+2(lnq/lnp)
n−2
·|x|
(lnq/lnp)
n−1
>|ρ|·|ρ|
2(1−lnq/lnp)
−1
·|x|
(lnq/lnp)
n−1
= |ρ|·p
−2
·|x|
(lnq/lnp)
n−1
>p
−3
·|ρx|
(lnq/lnp)
n−1
.
Ïd
3lnp>(lnq/lnp)
n−1
·ln|ρx|= c
(n−1)ln(lnq/lnp)/lnc
·ln|ρx|,
¤±
[3lnp]
lnc/ln(lnq/lnp)
>c
(n−1)
·[ln|ρx|]
lnc/ln(lnq/lnp)
.
a=lnc/ln(lnq/lnp),Ka>0…[3lnp]
a
>c
(n−1)
·[ln|ρx|]
a
.Šâ(3.4)Œ•, éx∈R,XJ
|ρx|>1,K
|dµ
ρ,D
(x)|·[ln|ρx|]
a
6[3lnp]
a
max{|dµ
ρ,D
(y)|: |ρy|61}<∞.
qϕsup
|ρx|>1
n
ln(3+|x|)
ln|ρx|
o
<∞…sup
|ρx|61
{|dµ
ρ,D
(x)·[ln(3+|x|)]
a
}<∞,ù¿›X
b:= sup
x∈R
{|dµ
ρ,D
(x)|·[ln(3+|x|)]
a
}<∞.(3.5)
½ny.
Ún3.2(i)D={d
0
,d
1
,d
2
}´n¢ê8, KZ(M
D
) 6=∅…=•3êk
1
,k
2
Ú¢ê
DOI:10.12677/pm.2022.121026223nØêÆ
ù[
a6= 0,¦{d
1
−d
0
,d
2
−d
0
}= {(3k
1
+1)a,(3k
2
+2)a}…gcd(3k
1
+1,3k
2
+2) = 1.
(ii)D={0,3k
1
+1,3k
2
+2},Ù¥k
1
,k
2
∈Z.XJgcd(3k
1
+1,3k
2
+2)=1,KZ(M
D
)=
±
1
3
+Z=
1
3
(Z\3Z).
y²(i) ŠâM
D
(x)½ÂŒ±wÑ
M
D
(x) =
1
3
(e
−2πid
0
x
+e
−2πid
1
x
+e
−2πid
2
x
) = 0,
⇔cos(2π(d
1
−d
0
)x)+cos(2π(d
2
−d
0
)x) = −1,
sin(2π(d
1
−d
0
)x) = −sin(2π(d
2
−d
0
)x),
⇔•3ên
1
,n
2
,¦(d
1
−d
0
)x= n
1
±
1
3
,(d
1
+d
2
−2d
0
)x= n
1
+n
2
,
⇔•3ên
1
,n
2
,¦{3(d
1
−d
0
)x,3(d
2
−d
0
)x}= {3n
1
+1,3n
2
+2}.
(3.6)
w,•3êk
1
,k
2
,¦
1
gcd(3n
1
+1,3n
2
+2)
{3n
1
+1,3n
2
+2}= {3k
1
+1,3k
2
+2}.
ùV«gcd(3k
1
+1,3k
2
+2) = 1.(i) =y.
(ii)-x∈Z(M
D
),K(3.6) V«•3ên
1
,n
2
Úl∈{−1,1},¦(3k
1
+ 1)x=n
1
+
l
3
,
(3k
2
+2)x= n
2
−
l
3
.Ïd
x−
l
3
=
n
1
−lk
1
3k
1
+1
=
n
2
−lk
2
3k
2
+2
.
Šâgcd(3k
1
+1,3k
2
+2) = 1Υ(3k
1
+1)|(n
1
−lk
1
)…(3k
2
+2)|(n
2
−lk
2
).ù¿›Xx∈±
1
3
+Z.
ÏdZ(M
D
) ⊆±
1
3
+Z.
x=
l
3
+ z∈±
1
3
+ Z,Ù¥z∈Z,l∈{−1,1}.K(3k
1
+ 1)x=(3k
1
+ 1)z+ lk
1
+
l
3
…
[(3k
1
+1)+(3k
2
+2)]x= (k
1
+k
2
+1)(l+3z).KŠâ(3.6)Œ•M
D
(x) = 0.ÏdZ(M
D
) ⊇±
1
3
+Z,
lZ(M
D
) = ±
1
3
+Z.
ŠâÚn3.2Œ•Z(M
D
)عk
1
,k
2
,lkeãíØ.
íØ3.3D={0,3k
1
+1,3k
2
+2}…gcd(3k
1
+1,3k
2
+2)=1,Ù¥k
1
,k
2
∈Z.XJ
C= {0,1,2},Kdµ
ρ,C
(x) = 0…=dµ
ρ,D
(x) = 0.
Ún3.4D={0,3k
1
+1,3k
2
+2}…gcd(3k
1
+1,3k
2
+2) = 1,Ù¥k
1
,k
2
∈Z.XJµ
ρ,D
´
ÌÿÝ,K•3êp,q,¦|ρ|=
q
3p
…gcd(q,3p) = 1.
y²Λ ´µ
ρ,D
Ì, ŠâíØ3.3 Œ•E
Λ
´L
2
(µ
ρ,C
) á8, Ù¥C={0,1,2}.2
¦^[4]¥½n1.2 Œ••3êp,q,r,¦|ρ|=(
q
3p
)
1/r
…gcd(q,3p)= 1.Ø”˜„5, b
r´¦(
q
3p
)
1/r
∈Q•ê, =?¿êk<r,(
q
3p
)
1/k
´Ãnê,K|ρ|•õ‘ª´
3px
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
∗···],Šâ
Ún2.4Œ•
(Λ−Λ)∩Z(M
ρD
) 6= ∅,(Λ−Λ)∩Z(M
ρ
2
D
) 6= ∅.
DOI:10.12677/pm.2022.121026224nØêÆ
ù[
Ïd,ÏLÚn3.2Œ••3λ
j
∈Λ(j= 0,1,2,3) Úz
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)
,˜•¡, éλ∈Λ\{λ
0
,λ
1
},kλ−λ
0
,λ−λ
1
∈Z(dµ
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
).
Ïd
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,Kρ´•§
(±1+3z
1
)x
n
1
−1
+(±1+3z
3
)x
n
1
−n
0
−(±1+3z
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
>0,06s
1
,s
2
<r.K(3.8) Ú
ρ= ±(
q
3p
)
1/r
V«•3êm
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´3px
r
−q…06s
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
)
!
,
2Šâ(3.7)1˜‡ªf,k
Λ−λ
0
⊂{0}∪
∞
[
n=0
ρ
−(1+nr)
Z(M
D
)
!
.(3.9)
éα6=β∈Λ\{λ
0
},ÏL(3.9) Œ••3ên
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
).qÏ•α−β∈Z(dµ
ρ,D
) =
S
∞
n=1
ρ
−n
Z(M
D
),l•3ên,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±aqr|(n−1).Ïd,α−β∈
S
∞
n=0
ρ
−(1+nr)
Z(M
D
),¤±
Λ−Λ ⊂{0}∪
∞
[
n=0
ρ
−(1+nr)
Z(M
D
)
!
.
ÏL(3.7)1‡ªfŒ•r= 1.Ïd, •3êp,q,¦|ρ|=
q
3p
…gcd(q,3p) = 1.
XÃAÏ`²,e©¥-D= {0,3k
1
+1,3k
2
+2},|ρ|=
q
3p
,Ù¥k
1
,k
2
∈Z,gcd(3k
1
+1,3k
2
+
DOI:10.12677/pm.2022.121026225nØêÆ
ù[
2) = 1,p,q´ê…gcd(q,3p)= 1.
Ún3.5XJΛ ´µ
ρ,D
Ì,Ù¥0 ∈Λ.K
(Λ−Λ)\{0}⊂
∞
[
n=0
(3p)
n
(Z\3Z)
3|ρ|
.(3.10)
y²Ï•µ= δ
ρD
∗[δ
ρ
2
D
∗δ
ρ
3
D
∗···∗δ
ρ
n
D
∗···],ldÚn2.4Œ•
(Λ−Λ)∩Z(M
ρD
) 6= ∅.
Ïd,ÏLÚn3.2Œ••3λ
0
,λ
1
∈ΛÚk
0
∈(Z\3Z),¦
λ
0
−λ
1
=
k
0
3|ρ|
.
Äk·‚y²Λ−λ
1
⊂{0}∪

S
∞
n=0
(3p)
n
(Z\3Z)
3|ρ|

.éλ∈Λ\{λ
0
,λ
1
},Šâ(2.2),(2.3) ÚÚn
2.4Œ••3êk
1
,k
2
∈(Z\3Z)Ún
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
.
qÏ•|ρ|=
q
3p
,lþ¡ªfdu
k
1
(3p)
n
1
−1
q
n
1
−1
−k
0
=
k
2
(3p)
n
2
−1
q
n
2
−1
.
·‚©n«œ/?Ø:
œ¹˜:n
1
>1…n
2
>1.Ï•gcd(q,3p) = 1,¤±þ¡ØªV«3p|k
0
,ù†k
0
∈(Z\3Z)gñ.
œ¹:n
1
>1 …n
2
= 1.K
k
1
(3p)
n
1
−1
q
n
1
−1
−k
0
= k
2
,l
k
1
q
n
1
−1
´ê. ù¿›X•3ên>0,¦
λ−λ
1
∈
(3p)
n
(Z\3Z)
3|ρ|
.
œ¹n:n
1
= 1.w,λ−λ
1
∈
(Z\3Z)
3|ρ|
.nþ=
Λ−λ
1
⊂{0}∪
∞
[
n=0
(3p)
n
(Z\3Z)
3|ρ|
!
.(3.11)
,·‚y²(3.10).éΛ\{λ
1
}¥ü‡ØÓƒλ=λ
1
+
(3p)
m
l
1
3|ρ|
,λ
0
=λ
1
+
(3p)
k
l
2
3|ρ|
,Ù¥
l
1
,l
2
∈(Z\3Z),m,k∈Z.ÏL(2.3) Œ••3ês>0Úl
3
∈(Z\3Z),¦
(3p)
m
l
1
3|ρ|
−
(3p)
k
l
2
3|ρ|
=
(3p)
s
l
3
3|ρ|q
s
.
DOI:10.12677/pm.2022.121026226nØêÆ
ù[
Ïdq
s
|l
3
,ù¿›Xλ−λ
0
áu(3.11)mý,l
(Λ−Λ) ⊂{0}∪
∞
[
n=0
(3p)
n
(Z\3Z)
3|ρ|
!
.

Ún3.6Λ ´µ
ρ,D
Ì, Ù¥0∈Λ.K•3êz
j
ÚΛ
j
⊂Z(j=0,1,2) …0∈Λ
j
,¦Λ
j
´µ
ρ,D
Ì…ΛkXe©)
Λ =
2
[
j=0
[
j+3z
j
3ρ
+ρ
−1
Λ
j
].(3.12)
y²ŠâÚn3.5 Œ•Λ ⊂
Z
3ρ
.éj∈{0,1,2},XJΛ∩
j+3Z
3ρ
6= ∅,K•3λ
j
=
j+3z
j
3ρ
∈Λ,¦
|
j+3z
j
3ρ
|= min{|λ|: λ∈Λ∩
j+3Z
3ρ
},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¿,=i6= j∈{0,1,2}ž, k[λ
j
+ρ
−1
Λ
j
]∩[λ
i
+ρ
−1
Λ
i
] = ∅.
ŠâÚn2.1Ú(3.14)Œ•,é?¿x∈R,·‚k
1 =
X
λ∈Λ
|dµ
ρ,D
(λ+x)|
2
=
2
X
j=0
X
γ
j
∈Λ
j
|dµ
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
|dµ
ρ,D
(
j+3z
j
3
+γ
j
+ρx)|
2
.
(3.15)
‰½j∈{0,1,2},eΛ
j
6= ∅,?γ
j
6= γ
0
j
∈Λ
j
,(3.13)L²λ
j
+ρ
−1
γ
j
6= λ
j
+ρ
−1
γ
0
j
∈Λ.Ïd
(λ
j
+ρ
−1
γ
j
)−(λ
j
+ρ
−1
γ
0
j
) ∈Z(dµ
ρ,D
),l•3êz∈(Z\3Z)Ún>0,¦(λ
j
+ρ
−1
γ
j
)−(λ
j
+
ρ
−1
γ
0
j
) =
z
3|ρ|
n
.Ïdγ
j
−γ
0
j
=
z
3|ρ|
n−1
,qÏ•γ
j
−γ
0
j
´ê,ln>1.ù¿›Xγ
j
−γ
0
j
∈Z(dµ
ρ,D
),
lE
Λ
j
´˜8½ö´L
2
(µ
ρ,D
)´8.qÏ•(D,{0,
1
3
,
2
3
})´ƒNé,ŠâÚn2.1 Ú(3.15)
Υ
1 =
2
X
j=0
|M
D
(
j
3
+ρx)|
2
X
γ
j
∈Λ
j
|dµ
ρ,D
(
j+3z
j
3
+γ
j
+ρx)|
2
6
2
X
j=0
|M
D
(
j
3
+ρx)|
2
= 1.
qÏ•A¤kx∈R,kM
D
(
j
3
+ρx) 6= 0,l
X
γ
j
∈Λ
j
|dµ
ρ,D
(
j+3z
j
3
+γ
j
+ρx)|
2
≡1,∀x∈R,j= 0,1,2.
DOI:10.12677/pm.2022.121026227nØêÆ
ù[
ŠâÚn2.1Œ•Λ
j
´µ
ρ,D
(j= 0,1,2) Ì,Úny.
y²½n1.1
¿©5: d^‡•|ρ|
−1
∈3N
+
…•3êk
1
,k
2
Ú¢êa6=0,¦{d
1
−d
0
,d
2
−d
0
}=
{(3k
1
+1)a,(3k
2
+2)a}.C ={0,3k
1
+1,3k
2
+2},K(C,{−
1
3
,0,
1
3
})´ƒNé.Ïd
(3p,C,{−p,0,p})´Hadamardné,Šâ[5]Œ•µ
ρ,C
´ÌÿÝ.•d·K2.5Œ•µ
ρ,D
´
ÌÿÝ.
7‡5:XJµ
ρ,D
´ÌÿÝ,K•3Œê8Λ ⊂R,¦(µ
ρ,D
,Λ)´Ìé,Ø”˜„5,·‚b
0∈Λ.KΛ\{0}⊂Z(dµ
ρ,D
),lZ(M
D
)6=∅.dÚn3.2 y²•3êk
1
,k
2
Ú¢êa6=0,
¦{d
1
−d
0
,d
2
−d
0
}= {(3k
1
+1)a,(3k
2
+2)a}…gcd(3k
1
+1,3k
2
+2) = 1.Ïd·K2.5y²
µ
ρ,C
´ÌÿÝ,Ù¥C= {0,3k
1
+1,3k
2
+2}.ù¿›X•I‡•ÄD= {0,3k
1
+1,3k
2
+2}
œ¹e|ρ|
−1
∈3N
+
¤á=Œ.D= {0,3k
1
+1,3k
2
+2 }ž,ŠâÚn3.4Œ••3êp,q,
¦|ρ|=
q
3p
…gcd(q,3p) = 1.Ún3.6 y²•3Œê80 ∈Λ ⊂Z,¦(µ
ρ,D
,Λ) ´Ìé.
•¦^‡y{y²|ρ|=
1
3p
,·‚I‡k‰˜OóŠ.
ŠâÚn3.6,ΛU©)¤
Λ =
2
[
j
0
=0
"
j
0
+3z
j
0
3ρ
+ρ
−1
Λ
j
0
#
,(3.16)
Ù¥z
j
0
∈Z,0∈Λ
j
0
⊂Z.
j
0
+3z
j
0
3ρ
=
p(j
0
+3z
j
0
)
q
∈Λ⊂Z.ù¿›Xρ
−1
Λ
j
0
∈Z,lΛ
j
0
⊂
qZ(j
0
=0,1,2).qÏ•0<q<3p…gcd(q,3p)=1,l•3j∈{0,1,2},n
j
∈Z,¦
j
0
+3z
j
0
q
=j+3n
j
.dÚn2.1 Œ•XJΛ
j
´µ
ρ,D
Ì, K−Λ
j
•´µ
ρ,D
Ì. Ïd•3ên
j
Ú
µ
ρ,D
ÌΓ
j
⊂qZ,Ù¥0 ∈Γ
j
,¦
Λ =
2
[
j=0

p(j+3n
j
)+
3p
q
Γ
j

.
?˜Ú,·‚Œ±ÀJm
j
∈Z,¦p(j+3m
j
) ∈Λ…
|p(j+3m
j
)|= min{|p(j+3n
j
)+
3p
q
γ|: γ∈Γ
j
}.
(3.17)
Óž•3µ
ρ,D
ÌΓ
j
⊂qZ,Ù¥0 ∈Γ
j
,¦
Λ =
2
[
j=0

p(j+3m
j
)+
3p
q
Γ
j

.
Ï•0∈Λ …0/∈
nh
p(1+3n
1
)+
3p
q
Γ
1
i
S
h
p(2+3n
2
)+
3p
q
Γ
2
io
,l0∈p(3n
0
)+
3p
q
Γ
0
,ù¿›
Xm
0
=0.Uìù‡öŠ,Œ±é˜ên
j
1
,···,j
n
Ú˜µ
ρ,D
ÌΓ
j
1
,···,j
n
(n>1),Ù¥
DOI:10.12677/pm.2022.121026228nØêÆ
ù[
0 ∈Γ
j
1
,···,j
n
,¦
Λ =
2
[
j
1
,···,j
n
=0
n
X
l=1
p(3p)
l−1
(j
l
+3n
j
1
,···,j
l
)+(
3p
q
)
n
Γ
j
1
,···,j
n
!
,n>0.
ƒA/,•3˜Xêm
j
1
,···,j
n
Ú˜µ
ρ,D
ÌΓ
j
1
,···,j
n
(n>1),Ù¥0 ∈Γ
j
1
,···,j
n
,¦
|
n
X
l=1
p(3p)
l−1
(j
l
+3m
j
1
,···,j
l
)|
=min
(
|
n
X
l=1
p(3p)
l−1
(j
l
+3n
j
1
,···,j
l
)+(
3p
q
)
n
γ|: γ∈Γ
j
1
,···,j
n
)
,n>0,
(3.18)
Λ =
2
[
j
1
,···,j
n
=0
n
X
l=1
p(3p)
l−1
(j
l
+3m
j
1
,···,j
l
)+(
3p
q
)
n
Γ
j
1
,···,j
n
!
,n>0(3.19)
…
j
l
= 0žm
j
1
,···,j
l
= 0,l= 1,2,3···.
‰½Sj
1
,···,j
n
,a
n
=
P
n
l=1
p(3p)
l−1
(j
l
+3m
j
1
,···,j
l
).Ï•
p(3p)
n−1
(j
n
+3n
j
1
,···,j
n
)+(
3p
q
)
n
Γ
j
1
,···,j
n
⊂(
3p
q
)
n−1
Γ
j
1
,···,j
n−1
,
l|a
n−1
|6|a
n
|,ù¿›Xa
n−1
†p(3p)
n−1
(j
n
+ 3m
j
1
,···,j
n
)KÒƒÓ.Ïdj
n
6=0ž,k
|a
n
|= |a
n−1
|+|p(3p)
n−1
(j
n
+3m
j
1
,···,j
n
)|>(3p)
n−1
.=
j
n
6= 0ž





n
X
l=1
p(3p)
l−1
(j
l
+3m
j
1
,···,j
l
)





>(3p)
n−1
,n= 1,2,···.(3.20)
-
V
n
=
(
n
X
l=1
p(3p)
l−1
(j
l
+3m
j
1
,···,j
l
) : j
1
,···,j
n
∈{0,1,2}
)
.
d(3.19)Ú0 ∈Γ
j
1
,···,j
n
(n>1) •
Λ =
∞
[
n=1
V
n
,V
n
⊂V
n+1
.
5¿j
l
=0žm
j
1
,···,j
l
=0(l>1),IOíŒV
n
−V
n
⊂{0}∪Z(cµ
n
),Ù¥
µ
n
= δ
ρD
∗δ
ρ
2
D
∗···∗δ
ρ
n
D
.ù¿›XV
n
´L
2
(µ
n
)8,ldÚn2.1Œ•
X
λ∈V
n
|cµ
n
(t+λ)|
2
61,t∈R.(3.21)
DOI:10.12677/pm.2022.121026229nØêÆ
ù[
e5y²q= 1.‡y{, b1 <q<3p.ÀJêN>a
−1
,Ù¥ad½n3.1‰Ñ.
Q
k
(t) :=
X
λ∈V
k
N
|dµ
ρ,D
(t+λ)|
2
,k= 1,2,3,···.(3.22)
Két∈(−(3p)
2
,(3p)
2
),k
Q
k+1
(t)−Q
k
(t) =
X
λ∈V
(k+1)
N
\V
k
N
|dµ
ρ,D
(t+λ)|
2
=
X
λ∈V
(k+1)
N
\V
k
N
(k+1)
N
Y
s=1
|M
D
(ρ
s
(t+λ))|
2
·|dµ
ρ,D
(ρ
(k+1)
N
(t+λ))|
2
6b
2
X
λ∈V
(k+1)
N
\V
k
N
(k+1)
N
Y
s=1
|M
D
(ρ
s
(t+λ))|
2
·(ln(1+|ρ
(k+1)
N
(t+λ)|))
−2a
6b
2
X
λ∈V
(k+1)
N
\V
k
N
(k+1)
N
Y
s=1
|M
D
(ρ
s
(t+λ))|
2
·(ln(1+|ρ
(k+1)
N
(3p)
k
N
−1
|))
−2a
6b
2


1−
X
λ∈V
k
N
(k+1)
N
Y
s=1
|M
D
(ρ
s
(t+λ))|
2


·(ln(1+|ρ
(k+1)
N
(3p)
k
N
−1
|))
−2a
.
1˜‡ª5g(3.22),1‡ª5g(2.1),1˜‡Øª5g½n3.1,1‡Øª5g
(3.20),•˜‡Øª5g(3.21),Ù¥bd(3.5)‰Ñ, =b=sup
x∈R
{|dµ
ρ,D
(x)|·[ln(3+|x|)]
a
}.3
e¡y²¥·‚•›t∈(−(3p)
2
,(3p)
2
).
d,·‚k
Q
k
(t) =
X
λ∈V
k
N
(k+1)
N
Y
s=1
|M
D
(ρ
s
(t+λ))|
2
·|dµ
ρ,D
(ρ
(k+1)
N
(t+λ))|
2
6
X
λ∈V
k
N
(k+1)
N
Y
s=1
|M
D
(ρ
s
(t+λ))|
2
.
Ïd
1−Q
k+1
(t) >[1−Q
k
(t)]·[1−b
2
(ln(1+|ρ
(k+1)
N
(3p)
k
N
−1
|))
−2a
].
(3.23)
qϕ
lim
k→∞
(ln(1+|ρ
(k+1)
N
(3p)
k
N
−1
|))
−2a
k
−2Na
=lim
k→∞

(k+1)
N
ln|ρ|+k
N
ln3p
k
N

−2a
= (lnq)
−2a
>0,
(3.24)
DOI:10.12677/pm.2022.121026230nØêÆ
ù[
w,•3n
0
>0,¦?¿k>n
0
,k[1−b
2
(ln(1+|ρ
(k+1)
N
(3p)
k
N
−1
|))
−2a
] >0.Šâ(3.23)Œ•
1−Q
K+1
(t) >[1−Q
n
(t)]
K
Y
k=n
[1−b
2
(ln(1+|ρ
(k+1)
N
(3p)
k
N
−1
|))
−2a
] >0,∀K>n>n
0
.(3.25)
ŠâbN>a
−1
,·‚k
P
∞
k=1
k
−2Na
<∞.Ïdlim
K→∞
Q
K
k=1
[1−k
−2Na
] Âñ˜‡¢
ê.Ïd,ÏL(3.24)·‚Uék
0
>n
0
>0,¦
+∞
Y
k=k
0
[1−b
2
(ln(1+|ρ
(k+1)
N
(3p)
k
N
−1
|))
−2a
] =: a
0
∈(0,1).
,,Šâ(µ
ρ,D
,Λ) ´ÌéÚ(3.25),ét∈(−(3p)
2
,(3p)
2
),k
0 = 1−
X
λ∈Λ
|dµ
ρ,D
(t+λ)|
2
=lim
K→∞
[1−Q
K+1
(t)] >a
0
[1−Q
k
0
(t)] >0.
Ïd,ét∈(−(3p)
2
,(3p)
2
),kQ
k
0
(t)=1.qϕQ
k
0
(t)Uòÿ•E²¡þ¼ê,lé
t∈R,Q
k
0
(t) ≡1.ÏdΛ
k
N
0
´Ì.´ù†Λ
k
N
0
´k•8gñ.½ny.
©·K2.5 y²Moran.ÿÝ'uêi8²£Ú ˜Ì5ØC5, ,Ún3.6 ò ÿ
Ý•¹"ƒ?¿Ì©)•n‡êÌ, X¦^½n3.1 |dµ
ρ,D
(x)|P~„ÝÚ‡y{
•(Ø.
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(11971109,11971190)"
ë•©z
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Analysis,266,343-354.https://doi.org/10.1016/j.jfa.2013.08.031
[2]Dai, X.R., He, X.G. and Lai, C.K. (2013) Spectral Property ofCantor Measureswith Consecu-
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[3]Dai,X.R.,He,X.G.andLau,K.S.(2014)OnSpectralN-BernoulliMeasures.Advancesin
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[4]Deng,Q.R.(2014)SpectralityofOneDimensionalSelf-SimilarMeasureswithConsecutive
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[5]Dutkay,D.,Haussermann,J.andLai,C.K.(2019)HadamardTriplesGenerateSelf-Affine
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ù[
SpectralMeasures.TransactionsoftheAmericanMathematicalSociety,371,1439-1481.
https://doi.org/10.1090/tran/7325
[6]Fu,Y.S.andWen,Z.X.(2017)SpectralityofInfiniteConvolutionswithThree-ElementDigit
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oreticProblem.JournalofFunctionalAnalysis,16,101-121.
https://doi.org/10.1016/0022-1236(74)90072-X
[8]He,L.andHe,X.G.(2017)OntheFourierOrthonormalBasesofCantor-MoranMeasures.
JournalofFunctionalAnalysis,272,1980-2004.https://doi.org/10.1016/j.jfa.2016.09.021
[9]Jorgensen,P.E.T.andPedersen,S.(1998)DenseAnalyticSubspacesinFractalL
2
-Spaces.
Journald.AnalyseMath´ematique,75,185-228.https://doi.org/10.1007/BF02788699
[10]Kolountzakis,M.N.andMatolcsi,M.(2004)ComplexHadamardMatricesandtheSpectral
SetConjecture.CollectaneaMathematica,57,281-291.
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DOI:10.12677/pm.2022.121026232nØêÆ

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