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PureMathematicsnØêÆ,2023,13(2),354-363
PublishedOnlineFebruary2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.132039
n‡êi8)¤MoranÿÝá8
•35
===xxx
4ŒÆêƆÚOÆ§4ï4²
ÂvFϵ2023c122F¶¹^Fϵ2023c221F¶uÙFϵ2023c228F
Á‡
bé?¿n≥1êp
n
>1…D
n
={0,a
n
,b
n
}⊂ZÙ¥a
n
<b
n
<p
n
"T©Ì‡ïÄd
êS{p
n
}
∞
n=1
Úêi8S{D
n
}
∞
n=1
)¤MoranÿÝ
µ:= δ
p
−1
1
{0,a
1
,b
1
}
∗δ
p
−1
1
p
−1
2
{0,a
2
,b
2
}
∗···∗δ
p
−1
1
p
−1
2
···p
−1
n
{0,a
n
,b
n
}
∗···
á•ê8•35§áòȵäká•ê8¿ ‡^‡§ù•Ed¼ê
˜mÌJøéÐg´"
'…c
•êħMoranÿݧÌÿÝ
TheExistenceofInfiniteOrthogonal
SetsofMoranMeasureswith
Three-ElementDigitSets
TingXiong
CollegeofMathematicsandStatistics,FujianNormalUniversity,FuzhouFujian
Received:Jan.22
nd
,2023;accepted:Feb.21
st
,2023;published:Feb.28
th
,2023
©ÙÚ^:=x.n‡êi8)¤MoranÿÝá8•35[J].nØêÆ,2023,13(2):354-363.
DOI:10.12677/pm.2023.132039
=x
Abstract
Forn≥1,let p
n
>1and D
n
= {0,a
n
,b
n
}⊂Z,where a
n
<b
n
<p
n
. Inthispaperwestudy
theexistenceofinfiniteorthogonalexponentialsetsofmoranmeasures
µ:= δ
p
−1
1
{0,a
1
,b
1
}
∗δ
p
−1
1
p
−1
2
{0,a
2
,b
2
}
∗···∗δ
p
−1
1
p
−1
2
···p
−1
n
{0,a
n
,b
n
}
∗···
which isgenerated by the sequenceof integers{p
n
}
∞
n=1
and the sequenceof number sets
{D
n
}
∞
n=1
.Weobtainthenecessaryandsufficientconditionsforinfiniteconvolutionµ
to have infiniteorthogonalexponentialsets,thisprovides a goodideaforconstructing
thespectrumofthisfunctionspace.
Keywords
ExponentialOrthogonalBasis,MoranMeasure,SpectralMeasure
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
µ•R
d
þäk;| BorelVÇÿÝ,XJ•3Œê8Λ⊂R
d
¦•ê¼êx
E
Λ
:= {e
−2πi<λ,x>
: λ∈Λ}¤L
2
(µ)IOÄ, K¡µ•ÌÿÝ,¿¡Λ•µÌ.eÌÿÝ
µ••›3;8ΩþLebesgueÿÝ,K¡Ω•Ì8.NÚ©Û˜‡Ä¯K´ïĵ۞•Ì
ÿݱ9§ÌäkÛ«/ª?'uÌïįK,Fuglede[1]u1974 cJÑͶÌ8ߎ:
Ì8ߎΩ •Ì8…=Ω ´²£Tile, =•3Œê¢ê8Γ ¦Ω⊕Γ=R
d
,Ù¥⊕L«
†Ú,Ò3ƒ˜‡Lebesgue"ÿ8¿Âe¤á.
dߎ®²y²3n‘9±þ‘êÑؤá, ù˜ßŽ3˜‘Ú‘þ´Ä¤áE´m¯
K. 'uÛÉÿÝÌïįK, JorgensenÚPedersen[2]u1998 cE1˜‡ÛÉÌÿÝ,
•´1˜‡©/ÌÿÝ. Óž•ÑØ '•óêž, IOCantorÿݵ
b{0,1}
•ÌÿÝ, Ø '
•
1
3
áBernoulliòÈÿÝdu•kk••ê.8,Ø´ÌÿÝ.ù˜uy-åïÄö
DOI:10.12677/pm.2023.132039355nØêÆ
=x
‚4Œ,, g•ÿÝÚMoranÿÝ&¢Œ€•–d‹ m, ŒþÌÿÝEÑ5, X[3–11].
·‚ •, äl Ñ8ÜΛ´Ä•ÿݵÌ, ̇l5Ú5ü‡•¡ y. Ïd, ·‚3
EÿݵÌž,ÄkI‡uÙ5. HuÚLau 3[12] ¥y²êi8•{0,1}gƒqÿ
ݵäká•ê8…=ÙØ '•
p
q
ng•Š, Ù¥p•Ûê, q•óê. ‘An, He
ÚLi3[4] ¥‰Ñ˜aáBernoulliòÈäká•ê8¿‡^‡.
Éþã©zéu,·‚F"í2œ¹•˜„Moran.ÿÝ.©Ì‡ïĘ‘œ/ed
ê8{D
n
}
∞
n=1
)¤˜aMoranÿÝá•ê8•35.{p
n
}
∞
n=1
•êS…
é?¿n≥1,
p
n
≥2,D
n
= {0,a
n
,b
n
}⊂Z,a
n
<b
n
<p
n
.(1.1)
XJ
∞
X
n=1
p
−1
1
p
−1
2
···p
−1
n
b
n
<∞,(1.2)
Šâ[13]Œ•,ÿÝS
µ
n
= δ
p
−1
1
{0,a
1
,b
1
}
∗δ
p
−1
1
p
−1
2
{0,a
2
,b
2
}
∗···∗δ
p
−1
1
p
−1
2
···p
−1
n
{0,a
n
,b
n
}
fÂñ˜‡äk•˜;| BorelVÇÿÝ
µ:= µ
{p
n
},{D
n
}
= δ
p
−1
1
{0,a
1
,b
1
}
∗δ
p
−1
1
p
−1
2
{0,a
2
,b
2
}
∗···,(1.3)
Ù¥, δ
E
=
1
#E
P
e∈E
δ
e
, #E•8ÜE³, …δ
e
•ü:e∈RþDiracÿÝ.·‚¡TÿÝ•
MoranÿÝ, …µ| 3XeCantor−Moran8þ:
T({p
n
},{D
n
}) =
∞
X
n=1
p
−1
1
p
−1
2
···p
−1
n
D
n
.(1.4)
3©¥,·‚í2¿[z[14]¥(Ø,̇ïÄdnêi8)¤Moranÿݵ3Ÿoœ¹
eká•ê8.é?¿n∈Z\{0},·‚½Âν
3
(n) = max{t: 3
t
|n},¿P
k
n
= ν
3
(p
1
p
2
···p
n
)−ν
3
(3gcd(a
n
,b
n
)).(1.5)
Äuþ¡PÒ,̇(ØXe:
½n1.1b{p
n
}
∞
n=1
,{D
n
}
∞
n=1
X(1.1) ¤½Â. K(1.3) ¤½Âáòȵäká•ê
8…=•3áêS{n
t
}
t≥1
¦{
a
n
t
gcd(a
n
t
,b
n
t
)
,
b
n
t
gcd(a
n
t
,b
n
t
)
}≡{1,−1}(mod3) …{k
n
t
}
t≥1
î‚4O.
˜„5`, ‰Ñ˜aMoranÿÝká•ê8¿‡^‡´˜‡(J¯œ, –8¤(
Jš~, Ù'…3u7‡5. ·‚ùŸ©Ùl&Ä":83 Ïf‡êÑu, éµ
{p
n
},{D
n
}
•
DOI:10.12677/pm.2023.132039356nØêÆ
=x
3á•ê8¿‡^‡.
©̇µeXe. ·‚31!¥Ì‡0˜~^óäÚ®•(J. 31n!¥,·‚|
^ ‡y{y²½n1.1 7‡5, ̇gŽ•":83 Ïf‡ êØOž, Šâp
n
>b
n
(n≥1)
¬íÑgñ,ly7‡5.ÏLEµ
{p
n
},{D
n
}
˜‡Ã¡•ê85y¿©5.
2.ý•£
3!¥,·‚ò0˜ÄVg9®•(J.
µ´Rþäk;| BorelVÇÿÝ,µFourierC†½Â•
bµ(ξ) =
Z
e
−2πiξx
dµ(x).(2.1)
f•Rþ¼ê, Pf(x) ":8•
Z(f) = {x: f(x) = 0}.(2.2)
éuŒêf8Λ ⊂R,Šâ5±9":8½Â,·‚k±e-‡5Ÿ. ù´EÿÝ8
~^•{.
Ún2.18ÜΛ ´ÿݵ8…=
(Λ−Λ)\{0}⊂Z(bµ).(2.3)
Ø”˜„5,©ob0 ∈Λ.
k•f8D⊂R,·‚¡
M
D
(x) =
1
#D
X
d∈D
e
2πidx
, x∈R(2.4)
•DMask¼ê. lµFp“C†Œ¤
bµ(x) =
∞
Y
n=1
M
D
n
(p
−1
1
p
−1
2
···p
−1
n
x).(2.5)
3.̇½ny²
é?¿n∈Z\{0},·‚½Âh(n) =
n
3
v
3
(n)
,=n¥š3 Ïf.K
n= 3
ν
3
(n)
h(n).(3.1)
3e©¥,é?¿n≥1,·‚Pµ
n
=δ
p
−1
1
D
1
∗···∗δ
p
−1
1
p
−1
2
···p
−1
n
D
n
,v
>n
=δ
p
−1
n+1
D
n+1
∗
DOI:10.12677/pm.2023.132039357nØêÆ
=x
δ
p
−1
n+1
p
−1
n+2
D
n+2
···.K
µ= µ
n
∗v
>n

1
p
1
···p
n

.(3.2)
Ún3.1D= {0,a,b}´nê8, Z(M
D
)d(2.2)¤½Â, K:
(i)Z(M
D
) 6= ∅…={
a
gcd(a,b)
,
b
gcd(a,b)
}≡{1,−1}(mod3).
(ii)eZ(M
D
) 6= ∅,KZ(M
D
) =
3Z±1
3gcd(a,b)
.
y²d(2.4), ·‚k
M
D
(x) =
1
3
(1+e
−2πiax
+e
−2πibx
),x∈R.
M
D
(x) = 0⇔e
−2πiax
+e
−2πibx
¢Ü•−1,JÜ•0 ,=
(
cos2πax+cos2πbx= −1
sin2πax+sin2πbx= 0,
⇔ax= l
1
±
1
3
,bx= l
2
∓
1
3
,l
1
,l
2
∈Z.
⇔a=
1
3x
(3l
1
±1),b=
1
3x
(3l
2
∓1),l
1
,l
2
∈Z,x6= 0.(3.3)
d(3.3)Œ•a,b3 Ïf‡êƒÓ.·‚P
a= gcd(a,b)a
0
,b= gcd(a,b)b
0
,a
0
,b
0
∈3Z±1.(3.4)
äó{
a
3
ν
3
(gcd(a,b))
,
b
3
ν
3
(gcd(a,b))
}≡{1,−1}(mod3)…={
a
gcd(a,b)
,
b
gcd(a,b)
}≡{1,−1}(mod3).
y²äód(3.1)Œ•gcd(a,b) = 3
ν
3
(gcd(a,b))
h(gcd(a,b)),Ø”h(gcd(a,b)) ∈3Z+1.
7‡5:{
a
3
ν
3
(gcd(a,b))
,
b
3
ν
3
(gcd(a,b))
}≡{1,−1}(mod3)ž,Ø”
a
3
ν
3
(gcd(a,b))
∈3Z+1,
b
3
ν
3
(gcd(a,b))
∈
3Z−1,·‚k
a
gcd(a,b)
=
a
3
ν
3
(gcd(a,b))
h(gcd(a,b))
∈3Z+1.
ÄK,e
a
3
ν
3
(gcd(a,b))
h(gcd(a,b))
∈3Z−1.dž•3z
1
,z
2
∈Z¦
a
3
ν
3
(gcd(a,b))
= (3z
1
+1)(3z
2
−1) ∈3Z−1.
ù†®•gñ,
a
gcd(a,b)
∈3Z+1.ÓnŒy
b
gcd(a,b)
∈3Z−1,äó7‡5y.
DOI:10.12677/pm.2023.132039358nØêÆ
=x
¿©5:†7‡5y²aq,{
a
gcd(a,b)
,
b
gcd(a,b)
}≡{1,−1}(mod3)ž,Ø”
a
gcd(a,b)
∈
3Z+1,
b
gcd(a,b)
∈3Z−1,N´
a
3
ν
3
(gcd(a,b))
=
a
gcd(a,b)
h(gcd(a,b)) ∈3Z+1.
¿…,
b
3
ν
3
(gcd(a,b))
∈3Z−1.nþ, äóy.
(i)däóŒ•,=yZ(M
D
) 6= ∅…={
a
3
ν
3
(gcd(a,b))
,
b
3
ν
3
(gcd(a,b))
}≡{1,−1}(mod3).
7‡5:d(3.3) Œ•
a
b
=
3l
1
±1
3l
2
∓1
,=
a+b= ±3(al
2
−bl
1
),(3.5)
k3|a+b.
(a)ea∈3Z±1,b∈3Z∓1,(Øw,¤á.
(b)e3|a,3 |b.d(3.5) Υa
0
+b
0
= ±3(a
0
l
2
−b
0
l
1
).éa
0
,b
0
?1aq/©ÛŒ
{
a
3
ν
3
(gcd(a,b))
,
b
3
ν
3
(gcd(a,b))
}≡{1,−1}(mod3).
¿©5:e{
a
3
ν
3
(gcd(a,b))
,
b
3
ν
3
(gcd(a,b))
}≡{1,−1}(mod3),K•3m
1
,m
2
∈Z¦
a
3
ν
3
(gcd(a,b))
= 3m
1
±1,
b
3
ν
3
(gcd(a,b))
= 3m
2
∓1.
Ïd,•‡l
1
= m
1
,l
2
= m
2
Òk
a
b
=
3m
1
±1
3m
2
∓1
=
3l
1
±1
3l
2
∓1
.
u´,oŒ±él
1
,l
2
∈Z÷v(3.3), Z(M
D
) 6= ∅.
(ii)d(3.3)!(3.4) Υ,
x∈Z(M
D
)⇔x∈
1
3a
(3Z±1)∩
1
3b
(3Z∓1) =
1
3gcd(a,b)

1
a
0
(3Z±1)∩
1
b
0
(3Z∓1)

.
ey
1
a
0
(3Z±1)∩
1
b
0
(3Z∓1) = 3Z±1,Ù¥gcd(a
0
,b
0
) = 1.dua
0
,b
0
∈3Z±1,K
3Z±1 ⊆
1
a
0
(3Z±1), 3Z±1 ⊆
1
b
0
(3Z∓1).
3Z±1 ⊆
1
a
0
(3Z±1)∩
1
b
0
(3Z∓1).,˜•¡,é?¿x/∈3Z±1,·‚kx/∈
1
a
0
(3Z±1).ÄK¬
•3z
1
,z
2
∈Z¦
a
0
(3z
1
) = 3z
1
±1.
ùw,ØŒU.3Z±1 ⊇
1
a
0
(3Z±1)∩
1
b
0
(3Z∓1).nþ,Z(M
D
) =
3Z±1
3gcd(a,b)
.
DOI:10.12677/pm.2023.132039359nØêÆ
=x
dþãÚn,·‚k
Z(bµ) =
∞
[
n=1
p
1
p
2
···p
n
3Z±1
3gcd(a
n
,b
n
)
=
∞
[
n=1
3
k
n
h(p
1
p
2
···p
n
)
h(gcd(a
n
,b
n
))
(3Z±1).(3.6)
y²½n1.1
7‡5:·‚Äky²•3áõ‡n≥2 ¦{
a
n
gcd(a
n
,b
n
)
,
b
n
gcd(a
n
,b
n
)
}≡{1,−1}(mod3).ÄK,
b•3N∈N¦n>Nž, {
a
n
gcd(a
n
,b
n
)
,
b
n
gcd(a
n
,b
n
)
}6= {1,−1}(mod3).dÚn3.1 (i)Œ•
Z

dv
>N

1
p
1
···p
N

= ∅.
bΛ´µá8…0 ∈Λ.ŠâÚn2.1(Ü(3.2), k
(Λ−Λ)\{0}⊆Z(bµ) = Z(bµ
N
).
ù†Λ•Ã¡8gñ,bؤá.
ùp·‚Ø”é?¿n≥1, k{
a
n
gcd(a
n
,b
n
)
,
b
n
gcd(a
n
,b
n
)
}≡{1,−1}(mod3).ey•3˜Ã¡
êS{n
t
}
t≥1
¦{k
n
t
}
t≥1
î‚4O.ÄK¬k±eü«œ/:
œ/I :{k
n
}
n≥1
Ãe.. Ø”{k
n
}
n≥1
î‚4~. bΛ´µá8…0 ∈Λ, dÚn
2.1(Ü(3.6)Œ•?λ
0
∈Λ,•3ên
0
±9t
0
∈3Z±1¦
λ
0
= 3
k
n
0
h(p
1
p
2
···p
n
0
)
h(gcd(a
n
0
,b
n
0
))
t
0
.
du{k
n
}
n≥1
î‚4~,•3áõên
j
>n
0
(j>0), ¦λ
j
∈Λ…
λ
j
∈3
k
n
j
h(p
1
p
2
···p
n
j
)
h(gcd(a
n
j
,b
n
j
))
(3Z±1),k
n
j
<k
n
0
.
Ø”•3t
j
∈3Z±1¦
λ
j
= 3
k
n
j
h(p
1
p
2
···p
n
j
)
h(gcd(a
n
j
,b
n
j
))
t
j
.
K
λ
0
−λ
j
=







3
k
n
0
h(p
1
p
2
···p
n
0
)
h(gcd(a
n
0
,b
n
0
))

t
0
−3
k
n
j
−k
n
0
h(gcd(a
n
0
,b
n
0
))
h(p
1
p
2
···p
n
0
)
h(p
1
p
2
···p
n
j
)
h(gcd(a
n
j
,b
n
j
))
t
j

3
k
n
j
h(p
1
p
2
···p
n
j
)
h(gcd(a
n
j
,b
n
j
))

−t
j
+3
k
n
0
−k
n
j
h(gcd(a
n
j
,b
n
j
))
h(p
1
p
2
···p
n
j
)
h(p
1
p
2
···p
n
0
)
h(gcd(a
n
0
,b
n
0
))
t
0

,
dk
n
j
<k
n
0
´•λ
0
−λ
j
/∈3
k
n
0
h(p
1
p
2
···p
n
0
)
h(gcd(a
n
0
,b
n
0
))
(3Z±1).Ïd,dΛ5Œ•λ
0
−λ
j
∈
DOI:10.12677/pm.2023.132039360nØêÆ
=x
3
k
n
j
h(p
1
p
2
···p
n
j
)
h(gcd(a
n
j
,b
n
j
))
(3Z±1).u´,
h(gcd(a
n
j
,b
n
j
))
h(p
1
p
2
···p
n
j
)
h(p
1
p
2
···p
n
0
)
h(gcd(a
n
0
,b
n
0
))
t
0
∈Z.
•Ò´`,
t
0
∈
h(p
n
0
+1
p
n
0
+2
···p
n
j
)h(gcd(a
n
0
,b
n
0
))
h(gcd(a
n
j
,b
n
j
))
Z.(3.7)
p
n
>b
n
(n≥1 ),¤±
p
1
p
2
···p
n
3gcd(a
n
,b
n
)
= 3
k
n
h(p
1
p
2
···p
n
)
h(gcd(a
n
,b
n
))
→+∞,n→+∞.
qϕ{k
n
}
n≥1
î‚4~.•k
h(p
1
p
2
···p
n
)
h(gcd(a
n
,b
n
))
→+∞,n→+∞.(3.8)
ù†•3t
0
∈3Z±1…÷v(3.7) gñ.
œ/II: {k
n
}
n≥1
ke..dž•3N∈N¦{k
n
}
n≥N
уӅ{
a
n
gcd(a
n
,b
n
)
,
b
n
gcd(a
n
,b
n
)
}≡
{1,−1}(mod3) .
XJΛ
0
(0 ∈Λ
0
)´v
>N

1
p
1
···p
N

á8,dÚn2.1 9Ún3.1(ii)Œ•?λ
0
∈Λ,•
3ên
0
>N±9q
0
∈3Z±1¦
λ
0
= 3
k
n
0
h(p
1
p
2
···p
n
0
)
h(gcd(a
n
0
,b
n
0
))
q
0
.
du{k
n
}
n≥N
уÓ,•3áõên
j
>n
0
(j>0) 9q
j
∈3Z±1,¦λ
j
∈Λ…
λ
j
= 3
k
n
j
h(p
1
p
2
···p
n
j
)
h(gcd(a
n
j
,b
n
j
))
q
j
,k
n
j
= k
n
0
.
u´
λ
0
−λ
j
=







3
k
n
0
h(p
1
p
2
···p
n
0
)
h(gcd(a
n
0
,b
n
0
))

q
0
−
h(gcd(a
n
0
,b
n
0
))
h(p
1
p
2
···p
n
0
)
h(p
1
p
2
···p
n
j
)
h(gcd(a
n
j
,b
n
j
))
q
j

3
k
n
j
h(p
1
p
2
···p
n
j
)
h(gcd(a
n
j
,b
n
j
))

−q
j
+
h(gcd(a
n
j
,b
n
j
))
h(p
1
p
2
···p
n
j
)
h(p
1
p
2
···p
n
0
)
h(gcd(a
n
0
,b
n
0
))
q
0

,
dun>Nž, {k
n
}
n≥N
уÓ. Ïd, (3.8)E,¤á. †œ/Iaq, ù†•3˜t
0
∈3Z±1
9áõt
j
∈3Z±1 (j>0)gñ.v
>N

1
p
1
···p
N

•kk••ê8, d(3.2) Œ•ù†®•µ
ká•ê8gñ.7‡5y.
¿©5: Šâ®•^‡, ·‚Ø”é?¿n≥1, k{
a
n
gcd(a
n
,b
n
)
,
b
n
gcd(a
n
,b
n
)
}≡{1,−1}(mod3)
DOI:10.12677/pm.2023.132039361nØêÆ
=x
…{k
n
}
n≥1
•î‚4OS.eEµ˜‡Ã¡•ê8.
P
Λ
σ
n
= {3
k
1
h(p
1
)3σ
1
+3
k
2
h(p
1
p
2
)σ
2
+···+3
k
n
h(p
1
p
2
···p
n
)σ
n
: σ
1
,···,σ
n
∈{0,1,−1}},
Λ
σ
=
∞
[
n=1
Λ
σ
n
.
?λ,λ
0
∈Λ
σ
…λ6=λ
0
.dž˜½¬•3êz,z
0
≥19σ
j
∈{0,1,−1}(1≤j≤z),σ
0
j
∈
{0,1,−1}(1 ≤j≤z
0
)¦
λ=
z
X
j=1
3
k
j
h(p
1
p
2
···p
j
)σ
j
, λ
0
=
z
0
X
j=1
3
k
j
h(p
1
p
2
···p
j
)σ
j
.
bs(s≥1) •1˜‡¦σ
s
6= σ
0
s
eI,K
λ−λ
0
=3
k
s
h(p
1
p
2
···p
s
)[(σ
s
−σ
0
s
)+3α] ∈3
k
s
h(p
1
p
2
···p
s
)(3Z±1),α∈Z.
Šâ(3.6)ªŒ•λ−λ
0
∈Z(bµ), dÚn2.1 Œ•Λ
σ
´µ˜‡Ã¡•ê8.½ny.
ë•©z
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