th
l − soshw−o t − hf af r−o e v − e rynon mp Xt y e − s t X ⊆ [zero − comma, ΠDX i s zero − P i h ard . Fo e − ach
o − ro − ever DQ ,th e u set s o
f Nwi hratio n a − ld
e nsiti es, s D2 (Π3) − o m p te.
In tr od uc t − ion . T e − hco l l c io nof “ Ω a tu ra−l y a i − r i n − g ” r
“ Ω o n ad o − hc00 s e
no
spe
ekno tedin
e
ptatnoles n−Σ0),arep of nayso ropeisrr et lrly@ra elyiv
smawnto ll.th
loca
be nrotp Baf ocacsmtedlab alνov mbthe erot hif d
atare n
m
o ral
oba
s et
o
wis
Π03c om p lee . A . D i ze n the n c n
u re
tha tif his wa s rue fo r ea c hb e n ≥2 , the n h eset 0 o f eal numb
rs a
e h
e
a e no r ma l to at le sto ne ba se n ≥ 2 s oul d be Σ4−c o m p let .C erta n
fo n
c
thi s e xa m pe i s n n−adh oc W e u dt hi s et e x teme l yd i f f i u tt o
m a n ag an d hen ce we aein c in d oa gee wi th D i te nsc o nj ct u r e T
h rei so m
e v den esupp o rt n th sco jnecru eN am ey e us as n [6] whic h sugg e
t h t n orm alt to b ase tw o andn o m l y t o b as t h e e ha v a w
eakf o rmo
in depe d nc . U n fo r una ey,s u h r o fsa re no cn sru civea ndth e co
e
jc ture app ea rst o b m o rn u mb e the or e ic t ha se t h o et ic I se me
r e sonab et o r pla ce t h es e tint he c ojectu re wi ht thee airt om ana
c olect on of s ubse s of N w ith den iy1/ n, for som e (va y i n )n ∈ N.H o w
e v ri nt hisc as et h lim it o n com pu t ei st h same fora l n, a d th s i sto
o i mpl W e t he n lo ok d a t th e su bse so f N wt h rti na l 0en ite
DQ, a 0dwere bl to ho w t a s p o perl yht e dif f re n c eof t wo Π3 set , i e
D2 Π3)− om pl ete.A s D Q i s at e as so me what nt u a lt hsis r th esur pr
19 1 M a he ma csS u bj c t Cla ific io: P im a r − y0 4 A 5Sec n da
ry 262, 11 K 1 6
j
ec
t
r
Π n(n
th
enD
(
o ∆ n+ s ets. I fX i s pr o pe r yD ξ( Πn), ten DX isp ro er y D ξΠ n +
o
1)s lo ng a s n ≥ 2.Hwev
e , o nh ed a ls de, a t th e n i e v es o f he d f f ere n c
h
˜ o r n =0 2,a n i ne rest n g p e nom no n ar es F o r < ω,
he a c
y wide
i
0is .
i f Xi p r ope l y Dm(2), t h n DX ip ro e r y Dmplus − one( Π3 )S ot he aalo y o f
Q t DQ e t ndst oa0lf i n e le ve ls of th edi ffe ren ce h ie a 0 c h, a nd no DX
s−ip o
ξ
˜
≥ ωa d0 Xis0 p op e yD ( Π2) ,the
ca nb p r − ope0 y wideD
m (Π3.I f
p er
0 DX∗r
˜ Dξ( Πα) an d Γ isth
Dξ(Π3F. or α ≥ 3, i f Γ = Πα , Σα, Dξ(Πα ), o r wide
wh
thenh−o
iclass
p r − lyere teΓ. I α nnparΓiscl replacera, weare dbya 1 + l − bα,
w
et o s
spro
sp
Xhatf o or − eyreverynΓonf f e − mDXp t
se t X ⊆ [0, 1, DX is
Π3 − h a d ; for each none m pty Π02
et
X ⊆ bracketlef t − zero, ], DX i
Π3−co pme te−semicolon f or ea c n ≥ 2, h c0 lec tion o fr el−a n u
en
mbes ta ta r o r ma
o rs m l n o m 0
o b a e iΠ 3c mlp e;0 a nd same n t ne d ab ve DQ
or DXo r a ny Σ2c m p et eset X,s D2Π 3−cm p et .
B otha uh or−s wou ldlike ot han kP r f e sorK echr sfo rhish l − e p , encou ag
e em nta n d en−g ag g − n c onver sat onsi nrea t − i ont ot h isw o period − k
No t tiona
n d back g o u ndinor m a t on . Fo se t − sA and
B , | A| s − ith
c a − ri−d nal−i y o A, A e − d n ote th to po og ial c los u − r e of A
e d not et he
tof a lf u ncti os ← ro m B in t A b A . If X ⊆ Aw
,
preima g
underf o f X by f ( X). W e soNme itm es i dent fy n ∈ N = {
N l e A< = n ∈ NAd eno e al f i nie seq u ncs o n A an d A N =
AN <∪ A .~0 ≤ aN d 1de n − o e th eco nst antz e roa n d co nst n t on ef u nt onsi n< 2.I
f
∈
A a n d n
∈
Nf n =
h(0 , .. , f parenlef t − n − 1) i a ndf o r
|
st ∈ A N, d eno e s t e en th of s( he u nq e n f r w h i ch s ∈ A,) s ⊆ t (t
e x t nd s s
m ea n s t s| = s a d sbtis th s equ en ce sfol o w edb y the se q − u e e t . W
us e R and Qt − oden oe−t ther ea l sa n d ra io nas,an d Pde o t − e the rr t − aio a
b wt ee n ez r an d o ne
Wed es c b t e o r h r a rc h n g t h s a n da r mo d r t rmn
o g o 0 A d i so na n d d f i n e th dif f ee n e h e arc h 0 o th e a m i uo us
cla ss s o s ∆ pac s eX , b sΣ 0(X)d−e e rn o e nt ec q oleti n fo Π opns ub ets o X
s , a o nd lΠ 0X
e lt 1
d t h l co f e s f 1 (
denot t he cl s d s
≥2 ,
Σ( X) =
A
{
vextendsingle
⊆ XvextendsinglevextendsingleA =
[
An, w here each
}
An ∈
Πβn(X
) and
β
n<
α
n∈
N
, Π1 = Cl ose s o 0
ors ech oc d − u a bl o r − o i an alpha − period or a ycou tabl or i alah nda nys qu
i
n e o − onef <ξ , Aλ = h β < Aβ( o hef se−q u e ces isc o n
t A = Dξ(Aβ iβ < ξ, th e di f fere n ce
o h ehA βiβ < ξby
x ∈ A ⇔ ∃β < ξx ∈ Aparenright − comma an t helar est su ch is ev n .
β
A co nt abl e r dn l β i seve n fw h0 nw e w i t eβ
= λ+ n, w it h
c
λ = 0 o r a l mt o rdin al n i s even . L tDξ (Πα) be t hec ol lti
o n o se t of t
r
h 0 fo m for hξAβ << ξh ede c rea s g <req ie
nti srg d n au t ) .
So
m
De n(Πos0)Π=Π e0 ,
(
DD2 (Π Π0alpha − zero =,)s {te)A−adBD(AΠ Bn0)i ∈st Π0 αe ,cole a n dAti ⊇ eB}ono(soists
ot nh R, [0, 12ef ) isa
ypica
, 3 α h 0
˜
wide
α
˜
Γ, 0 t he0 ua lc−l a s Γ be h eceoll ct o n o fc o m p e − lwide
F r an y cl ass o widest
˜ e ns
t−s
(
α
o s ei n Γ o D1 (Π) = Σ α, an dsa y A s p rpo ery Γ iA ∈ Γ −Γ We
n e e t h f o w0 g ee me n t ry f a ctsab o ut th e d f fere n hi ra rhy c
sa e − s.
T − he Dξ Πα ) e s r − a c s du er :
parenlef t − i) i nte rs cto s − n wi−t h Πα e − st s − semicolon
er−s e o − i n s w − i Σα s i ξ s e semicolon − n
˜
( iv ) union s w with Σα 0 s tsi f ξ ≥ i ωo. wide
0
E a ch of the sem pls adua lprop e ry f r th e Dξ(Π α)sets. F or e xa mple
˜
s − ays th @ht e wideD
elo
ξ Πα) e t s r
bi n ni g the abo v pr o et s w i
< α), th en both A− B n d A ∪ Bar e Πα
=β a n d ξ ≥ ω ) :
( v ) he Dξ(Πα)et s are cl s de un
wil−l n ee d thi o − f r β = 3 late r − period
In o der t ode e − rmn−i e t he ex act l ocat i − o no fa eti nt e ab e − v h
ie ar hy
one m us t p r − od u ce a n u pper b un d , or pro v em e m b e sh pi n t e l − cas s Γ,
˜ .I ngen e ral h e ow r o un d
an t h en aow er b u d s o w − i g h e e t sn o in wideΓ
i
a r−e m o e d
f o be in Γ e tehr.T h ar ep rop r ,n al lt
0fa Plio hto po o icas pN c
X isca led Γ−hard ( f rΓ = D ξΠαN orD
h c l s es
ξ(Π
)) if or ev
f
f
on
X
erei
tht−h
@s, a← f ( nA) uou=C. T h nctuif n,A :2 s − iΓ− → har d ,
bei n g Γ−ha r d,
ge [7] (s − ui n
A i sa so i n Γ, w
A
is
it−an−g
ibse B weena⊆X,
e
e s ay
be−tes,
rm n
e in ac
el diet f f er
rn
y
cc
o P ol shspa
ass ret
ha tth
es
x∈
C⇒
f ()∈
[3)
C sh
dthb
oee−w
A
a
nd
t
B
cnti n u o − usf nc to n
A,
ad
x 6∈ C
⇒
above.A 0 u be twide
˜ A o
∈
eyC
6∈
Cta
uhs,th n
Γ.If
⇔n
( )∈
f ad dit−i onA
t
A∈
˜
twidex
Γ− c − o
mle e.
deit−e m e
ubs y
o
iΓnj oze rp−m
s
t
o−i n
:X→Y h
f (x)
Γ(2
∈B
Wa d
e−l
Cor
s oY
P
sh
proet
nio nal , oΓ
pe
spac e L −
re
.
If B = ¬A = Y − A, wew ite C ≤ A r C ≤ (A; ¬ A, an d
say C i s W
adg eedu cb e to A. W a dge ’ resut men io e daW bov e w ast a
tor a B re l su bet s A a nd B f er - i mens ional lP toish sp a c s , h re A f≤ B
o r ¬ B ≤ s A. L o u v au ano dSai t - Ra ym n d 2 a r how e e
e o t n d u s n − gan y i − t c t r m − ic − ay − parenright f o C ≤ W
c l − o ed g m − aes I − t m pi st at fr e h − cl s
o
Γ = xi − parenlef tΠ α r Dξ( Πα)(α ≥ 2
o
t h − er eis a Γ
m e s HΓ ⊆2 N su ch tha t for l − a dijo nan aly t c
−c p t e
A an B ( i na ny P l − i hs p c ) , eihe r HΓ ≤W
( AB )( b y ao
A ⊆ S and
ne to − one o − cn − ti u − o f u c on or t hreis a Γse t Ssa−t
hth
uc
B∩ ⇒ ]=∅.T J∝F r F o ∞r − owner ← l s ses (s − in e − cw e on yl w − o rkin
n
P oli sh s p ac e), ebing Γ −ha d − rad be
ga n no −
i
ry Γ
n
t
B
t ing
˜
wideetN
oaice athe
st samehatif hC s .Γ Hehyphen−h a ced aseand wiC llbe≤W ro p, then
C ≤W
(AB
), a n d Dis a ny s tc not ni−n g A
≤ WD
Subs
asi f ac ts
l nu m
mpl e
.
sΓ−
mp
e.
c o
−har d. A e−lndi f
B f f itsi Γ
an d dsj o n − it fro m B, then
C
tso f N withg v end e ns i te s . W e d e c ib e h e − r e t he b
n d p r p e r esa b o u t he de n s t−ie so f u e−st o−f hena u−tar
b er ,t h t w − e ne T h s − i t p c c ov re d in d eta li n [1] o ex a
.
Dei − ni
T h u, wh enev e it e i ss δ(A)
T { r ← u.nmapsto−inf inity ← 0
[0, 1 ∩⇒, e ]T J ∝F3
X=
∈ R ∩, ⇒ ]T aJ∝ i F Oo← ∃ h .∃tnegationslash−arrowlef t
← cu ren c so A i nN F orn o n mptyX ⊆
{A ⊆
N|δ (A) ∈
X}
= {r} w e w ite Dr fo r D{r}parenright − period L et
h e c l l − e c ion of
DE = D[0, 1] e − d n ote
(
h
0 i f n topo nl gy ) For ss∈ 2u<bN,l etf k ts k e=| n{i ∈
1u} an podl uet |
d en ot t − h e l e g o s. W ec
δ(s) = bar − s − s
s pD−o m s( ) ts(it=
k
.
|
Q ∩ comma − arrowdblrightunionmulti − bracketrightT
o r α ∈ 2Nthed e s tyof α ex its ifft he s e − q uen ce {(α )}n N co v − n ergs ,
F nwhich c a eth e m t o f the s q s ne c e i t edensi y o f
α T i s how sth a t DE and D0 sareΠ l 0 , i n e u s .
(
| δparenlef t−intersectiondisplayelement−alphaN } )T
]
−,
circledottext − parenleftharpoonupright−bac
w<1 n, wn i h Ni clo intersectiondisplay − elementtpbracerightBigg − Nn unionmultidisplay − elementN
˜ n (\ he bracketlefttp
os ebracerighttp(bracketleftex/d bracerighttpbracketrightexp( ).wider
f
u N − t − parenrighttpre ( e \bracketlefttp ill / o (t
bo h / err writ α ng numbe r ( qua tif ire sas nco unt bleinI t se c t n uso rcoun able
no n , n or ill ev e ify h
t s l − e a h − t at h e − y ar .D0 is a s Π3 snc e
α
e δα
∈
⇔ ∀n ∃N ∀k ≥ N (δ(k ) <
D
n − parenright
N − lmn≈
1/n).
v r − ycl e − s t o b
henδ(αn)a
)
(α
) − δα
|δ
| <
1
2
n ( n+1
(
t f I = m in f {(α ), a n d +S = lim
th enforeve r y r − eal n umb e r r ∈ [I, S], r is al i i − mtp o − in tor c u − lste
rvalueo
the s q ue nce {δ α )}n ∈ N.
o r pr o d c i ng sub
s s tiw h
gvi e two mth o d s of pr o duc n α ∈ 2N o t a t he de n f α i s e a
o
to com pute . Th e firs inv o v c o y n gt h v la u es
f as e
S
q enc {x nn ∈
wi th xn ∈ (0, 1) T hei d
,an d δα k +1 i s beteen δ(n)an
d δ α n+) w eneve r k is be twe | αn |an |
αn + one − bar. Thu s t e − h dens y o α wl ei s tif f t − he s eu e n e
{x n}n ∈
c onv rgs, ndt h − e l i mi to f th s eq ue n ew l bð e d e si tyof α. G iv en
a
k
k αn
k +k
k
(k e x s ktssn ce {δα( nb1 }k ∈
nb1 .I δα n) ≥ xn+ , f i t h e l a
|
δ( nb0 k )k=
bk
a n d se α n+ = αn0 . L e
αn|+ 1 + 1 fo r a l n ∈ N U
| xn
1
Nt r@s@ δ(α n) an d incea
t k ∈ N− {0 }su c ht h t
n|
αα+k
≤x
n+
t α = n ∈ Nαn ∈ 2NCl eary , | αn +1
in th m i n m a t o k a nd ( ) , w
1)
−δ (+| |<
≥|
alpha − n |
be we e en delta − parenlef tα)ano δ l (n+ d ) hw ee−n ve r m i bn e ween s|α n | and
|
h
n+1| S o h
α
e n s tyo f α e i ts f f αthe squ enceof p a ia d eni te s f α is Cauc
e{ x
} ∈ f f,
n vege s ubseq u n hu
l − i m xnk = l m δ(αn k).
k → ∞ k →∞
T hi sg v e t hen a c a oni al w ayto p r d u c e α w ith δ(α) = , r f
o − ran yr ∈ 0,1 N otice a s oth t i xnh a p p e − ne dot be zero or o n comma − e w
e co dl r − epla e − cxnw i
1/n or 1 − 1/n respec tv e , y and hence we c a nr
ioni nvo e p rat t ioni n g
i − nf inite N o l − lec t onof se ts wi h po sitve d n si e s a n d lp ac ng ac o p y fs
o m α n ∈ 2 o n t h e nt hseti n t e pa ttio . The ev e n wh e n h
e pa it i o i s nfi i t o n e c a n b asic aly a d thed es tes Ing e ner lt h s is
not r u
snce th eu ni n o
s
ad ensiy z ero . B u tw h e nt he pece b ei n g c o mb n ead re c onta
s
t
n edi dji oi nt set s wi t h p os tv e den ii e e ver yth ng w ork s o t f i ne
L etSI ⊆
a nd {A}n∈ I b a f am i yo f pP
ai rw i disj ntsu b ets o fN su
h
A
a
c
tha n ∈ I
n = N( i . a ap r tit o n
N N) fore
chn ∈ I ,the d e nist yo f
Anex i
and isp ost iv a ndl mN → ∞ n = 0δAn) = 1 Fo r ac h n ∈ I, et αn ∈
2 b such th at δ(α n − parenright e i s − t.D ef in eC ⊆ N ,t h e eto
o
batin e
db
pla yig n
a
op o
f
α n o nA n, as oll w :
Fi s ,s i − n ce (A ) > 0, A n sinfinite Le t {a }k ∈ N b a
n − o eto - nein
crea s − ig e u − n m e − ra t on n o f A n T he n for e c − a h m ∈ N th erea re
un qu e n
an
kus
αn
k)
=1 .I
t
h
th
at m
= akW e
p − ut m
∈C
if f m
=
ak a d
coue rset we can s t l e mak e rht
e dv er e − n dsn−e i t e s, t h
neo
esiy o C does no ex i t .
S om e 0Π0− c o mp l − et e
se s. I n t his sect o n w e e s abls ha
s otrn g re d u ti o of a Π3 −c opm l tese t to t hese t D0 T hu swe ha0 e an a
f f i − r ma t ea ns w t o aq ue s − ton of K e his , wh o a k e dfi D0w a s Π 3 c
o m pete . O ncet hsi do n e, w e a eab et o hso w
C3 = bracelef t − beta ∈ NN | ∀n, β ← (n )s−i f i n i e } = {β ∈ NN| li→ mi f β(n) =
0
i Π 3− c o m
T he ore m
3 C
DE ar
n
∞}
∞
p e t − e(se ef or e am pl e( 24)n[ 4 fo ap roo )
≤W (D0; ¬ DE). Inpar t − i cu ar , b th
D0
a nd
0− comp et e − period 3
e
r
P ro of . he s e − con dp a t − r o f the th em f l − l o w sf o − r m t − h e f i be
r
t
a se⊆ quence {xn}n ∈ ENs othat= xn d ¬ e T pn e ds eon y o na s sf i
β. W eth n ro udc he c a o n − i c a α wit hin
put {xn} element − nN Th f u − n tio
Π
7→
α
(f rom
β
α d e p don lyo n t ef i r t N va l uesof { n}n∈
f in i in al s gm t o β. h s qen exn = 1/
mu stf irst f i x β ot at β( n ≥ ,and β
n t (sotha l m →∞ 1/β n)exist i f fi i {rszeo). F rβ ∈
by
β0 n =
β
(n2) + 2
i−f
N w h h de pe n onlyo n
(n)al mo tw o k b uw
sn ot N v n a lly0 oc ns tN
N , d fi n e ∈ N
niseven;
() n + 1 i f nis odd .
Th e n β 7→ β0 iscn n u o san d β ∈ C ⇔ 0 ∈ C . G ven β ∈ NN,
l e t α ∈ 2 b et here ul o fr unni g u th ce a n nia l o str cton on n pu t
{1/β0n }
Thnt h esequ n c e {δ( )} oa way s o au i sa su si u n ew
c β (2 + 1) = 2n + 2 He n c e thede
β∈C⇔
β0 ∈
C
⇔
im
1
=⇔ δ
α) = 0
3
Thi
sh ws
C3 ≤
W
(D; ¬D
⇔α∈D
3
E) and co mp l − e tes
⇔
n → ∞ β(n)
t e proof .
∈E .
α
0
0
h
D
th
de
n − o t eth ec
n t i n u o ebi−t s
wth−c i g
{
φ(
=
(n
)
1
0
rph i − s m
hom om
of 2
if α(n
if α() =
0.
Th e n g shows C3 ≤W D. Fina y , i X ⊆ parenlef t − zero, 1), e − l t
x ∈ X be rb i − t rar period − y L e A ⊆ N have d nsiy x. F ix n ∈ N su c − h t
a − h x + 1/n < 1. L e − taA 1 bedi j in
fr om A0 a n d h veden s − i ty 1/n. Gi en
i heresuot f p
δ(C) w ex s one − i
f f C )exst , a
β∈
1
a nd
x
C3 ⇔
δC( )ei−s ts
n He c − e 3 ≤
DX∩ ⇒ D -
W
⇔ δ(C1) = 0 ⇔
δ(
α) = x + 1 · 0
(Dsemicolon − x¬DE, so
3≤
W
⇔
α
∈ Dx .
DXbec aus x
∈X
= ∅.
t − h e eq e − u c e {d}i ∈ N ∈ nN
w ri ey ∈ n S N, ffo r ea ch k= 01.
a x ∈ [01] s norma lto b s − e
e ach s ∈
P
suc h ht a x =
i = dibn , a d d 6= n− and f i
, n ay −1, s m p
n o ma o b ae
n, a ndw r e x ∈ N, iff o each m ∈ N a
nm + 1, t n r
m+
δ(
in
{∈ N |i
= s(0 ), d + 1 = s 1
1
d+m
ase nfn tebasenex
f r − e q e nc y , e at c Itiskn ow nt h a t hes eo fn u m bs n k[0, 1 tha t r n orm
toa l b ss n ≥
5[bracketright − parenright I ssS
r − ta − i g h tf or w d − r t o se e t h atSN n a nd N nar e Π0 si n
D1/n s Π 3. On e of e m i n q ue t io 0 tha t mo iva et d h i s u d y wa st otr
e
o h w that NN = n ∈ NN n was Σ 4 c o mp et . W e w r euna 0 e t o a
nswe th is,bu t d id ma n getos h o w h at N n a d SN n a reea c h
Π3− o m let . A
N n ⊆ SN n t e olo wing re ul sho ws b o tho fh es s m u a n e ous y
T he o re m f ive−period Fo reach n ∈ N−{0, 1}, D0 ≤ W (N semicolon−
en. Lt{ik}∈N b anncrasg
n¬ SN parenright−period I
npart cula bot
∞= d/nibe
SNn
b as Pr oof.Le−t x =
x e i dnon mberof ththateet iI0={i n ormalt∈ N e k i e u r t
nd
Nn
otherw i − s e . a t
P
0
,
1
x=
7→
x0
s
n
i
i
otnu
ou.If α) ∈ D
,
x
en
s
t coo nne s
T
aiyi
c − ha
r
an
h
A dn i αDeh bs e tne
e
Bo r − e l c l s ses ofD X. W e o − nw
tu rn t ot ep r o em
of cl
l
t heBo relca
s o f DX i n t e m sof t h cl aso f X. T h fac tthat s
c
ne x
ea ti onshp ex tsis su prs−i n g B a cal y DX h s n − oem or equ a nt f i r
le s o nthes am e ide oft h e i ea r ch as X. We ta rtw i t ht u
Π03 omple e .
Pro of . L t {U } bea co unt
T able as so ope nse s or R ∩, 1] with
F th e usualto poe ogy .n Let X N =
G b e ay non m p y Π0s − u b set
⇒o [, 1] whe re ach G i sope n . A m om ent ’ s efl ct on h ow tth a t 2
f
∈D
X⇔
α∈
DE
and∀
k∃n∃∀p ≥ m
( δ( ∈ Un ⊆
U
n⊆ Gk .
α
s
h
n c m e mbe sh p N( , n, p) = {α ∈ 2 | δ (αp) ∈U n⊆ U n ⊆ G
com p etl yde t min d b α pan dw he t er or no t U n ⊆ G3 k w hi
inde 0 en d nt of α p) N k np)is c 0open fo eac( k n
y P parenlef t − X) h − t s e o α ∈ 2N s ch ta t 3 m et e F u r r , e
t
)
e tX
hnCPr (o Xl a sr−y Π7.L
03andα
X
[0⊆
∈D ,1
⇔α
∈
).
D
Π
e
∀k∃n∃muniversal − p ≥
m
⇒(
ben1 DpE∩mpt y. XT.J
∞ 3 ← ∃ . ∃negationslash − a
T {← ∀ .
∝F
parenlef t − i)f − I X is Σ2, h e − n D X i s
D2
i − s Πα (α )f r α ≥ 3, th e n DX s − iΠ1 +α ( Σ1
˜
i )I f X sD ξ wideΠα0
r
n
ξ ≥, h e nD X
(Π0
we
o
α
˜ D m( Π0 fo
( v ) I f X i wide
m
2 D 1 Π .
≥
< ω, t h e
δ( ) ∈
U
n⊆U
n
⊆
α
DW
=
D−
D
X,
D
CeF ∞ly ← .W
∃arrowlef t−negationslash=T
[
] \K
X, an−d DZ =∈
\
D
F
T
Y =
bracerighttp
An easyin d uct on
th n0 s h − o w s , r∈ n ≥ 2, t hatf r e ch
s
Π n Σ )e t X ⊆ [zero − comma1], t h r is a Π plus − n1( Σ n − zero + 1)
set P X) ⊆ 2N s c − u hthat
9)
α
∈D
X⇔
α
∈
DE ∩
(
. ∝ F ∞ 3← ∃
X
. ∃arrowlef t − negationslash{←
∀.
(
o se dr tn e − re−s i o − ns w
f(α ) = 2α o th e r w i − se, s
h Π30
s − tT he
ct i − o
f : 2N
→ Ri−gv e
←
, 1bracketright − comma t e n D i s Π0α r Σ40 α.
+α = 1+ α = α, s o(i h l s (a so t h − e e vels o the pr o c − eti v − e h
e − i a ch y
o no t i cr eas e f − ro m X to DX ) Us in g (parenright − eight a nd (9),o
n e sh s t hat o r
thα a d ξ≥ 2 th
s ( ifoll w s , s i ce D (Π0 ) se t − s ar ecl o e du n rin e ec o n wit h
d
≥ 2, t het n sD =
se t s ( as o g snα ≥). f 1+X Dwide
˜ (Π0)or α andξ
lna 2 I s ξ α 0 0 X
∩⇒ D - ¬ owner − negationslashX⇐w¬hc h iste i er e − ci on of a Π3 t
˜
w t h a Dwideξ
Π1+α set .
0set
e ξ (Π + α) i s
α ≥ sec−t 3, os−n o − rwiαh Π= a2n
d ξ ≥ ω, t e cl s − a s D
dun r
l e − swide
˜ 0e
by i( ) a d thed e
so(v h o ds .
T eup p r ou nd of D (Π0) f or X = Q, t er iona s tu rs ou t o b
a w er o u nd l o . T sis r th r s rpr ing , hnc every f
, w e tsar kn tow t o be prope r − l y l a − c te d a b vet hethi rdle vel oft h e Borel
hi r − a rc y . Wen o w
sho wt−h ta DQ s Σ0har . It u − t rn o u t th a − t n o DX is Σ0 c o m
l − pe−te( ec−x e p
o f co u r − s e for X =
3−o m plet b y T e o em 3, a n d 0 p r of th isf ac t wlb e th e se cn haf o th
c
epoo0
at DQ i s D2( Π 3)− omp et e . O ur r d c ti on her euss
n on san da rd Σ3
S3= {α ∈
2N ×N|
− co m p lees e t, n am e y,
∃R∀r
≥ R∃c(alpha−parenlef t r, ) = 1)}.
N 7→
α∈
2
n×
t his ma ke t he nu m e r fr osw i n α wt ho u a ny o nse qu alt ot hel m
in o β,so βelement − negationslash C3 ⇔ min n→ ∞ βn ) i f i ne ⇔ α ∈ S3.
t
We d e f inei nd utv e α ×n fro m β n so t ha β 7→ α is c nnt u ou s a n a
ts tag e n we mu
defi n e α en re nc o m nn f r h row s 0, 12, ... n− 1, as wel l a
n, col mu ns 0, 1,. ., n) a s flo o w s :
Sta g e 0 : If βparenlef t − zero) = 0, se αzero − parenlef t0) = 1 and i −
f β() > 0, s et α(comma−zero0) = 0 T hu s Z, t e n u−mber o a−quotedbllef t
llz e r ” r o s n α1 × 1 is e − i ther 0 = β(0) o
1 ≤ β (0), n − d h e nc Z1 ≤ β zero − parenlef t
S a g e n : W ea e gien α an d β(n), a n d m u t e f i n et e f i st n+
e trie , in b t h c o u m n n an dr ×o
w s i n αharpoonupright − n × n t ha t ra a l z o − r s .
I f β( ) ≤ Zn, e xt nd the “ 014 s − beta − tquotedblright − parenlef tn)
00
“azero
o w s of α n n y a d in
a eroi nc o − l umn n al t e − hrem
o n e n ol u − mn n; and d e f i e e f i st n + 1 tri e i nro w n to
b e n e Her e, “014r s ” i sdef i e df r − o m h e n d c sof he rw s ,s oth ers t 5ro ws
r tot he 5 r w wth low e st indice
I f β(n) >Zn , e xte n d ev e r − y “ a l z e r ” ro w by ddn g a z e
no l um n n
very o w hat lr e ad yh asa o en g ts a o ne i nc ou m n n;an d
m ake ow n be gn wthn + 1 z − eos . H n ce Zn +, h e nu m b e ro f “ l z e00o r
ows
α (n +1) × (n + 1,i sei t er β(n − parenright or 1+ Zn ≤ β(n), a
da ai n Zn + 1 ≤ β(n M or p r es ey , f r r < n, le t Z
it h n d e x r rth{ a are a l z e os . T hen fo r < n w es e
α (rn) = 0 if ∀ < n[α(rc) = 0] a nd Znparenlef t − r) < β(n;
1 o herwi e.
A n d f o r c ≤ n s t { 0i − f β(n) > Zn;
α(n, c) = 1 o t − h e r w s .
m
k
(
f i rs t k “ a o − l − raz n≥
n) ≥
th en Zm ≥ ko − f r a l e
o00 ro Nw ,in
,
c
T h u,
n) ≤ t h n m er of “ l − le − zr o
TT h en β e take s n the va t e tk in finte my f t n β.L t n a < .. e < m nβ b e a n cl
ect on o f +1 nat u alnu m e r s . W esh o wthat 1 r som. e i =k +1 o k + 1
row
m
βelement − negationslash C3 ⇔
S a n ∈ ¬Σ0 . ≤
S3
W
So S i
3
h
α∈
S3 ,
d a n d it i sst a i g ht orwa d t o se e
th
3
3
Pr op o s − it o − i n 10.S3 ≤ W (DQ; DP
b − e a p t − rt − i n of
e x mp on ca ta ke {2 np + 1 − 1p} ∈ N
∈N b a
n{k}
i n es
aparitio no (N
su at b − lLo−f
r
o − ur =s
(
=
{1iα
r0) = 0 f or
r
A
n) .L et
con dcano i − nc olc ons tructi
∈ 2N × N,
α∗
f or
c 0
l − et
a − ll c0
≤c
ote − rw s − ie .
Th n α 7→ {∗}r ∈ Nisc o n t − iν u − o s f (o − m2multiply − N Nt (2N )N) and
α i − s evntual z e o ( h en ce (αasteriskmath − parenright = 0) i f f row r o α
has ao e − n, and αr iside n t − ic l − l yon h nc e δ(α∗
na
g α r o B, o r r ∈ N, ∞P
d 0on B ∗. Th e n
∗
δ(f (α))=
r!2r + 1
r=0
(f or w hi − cexa h alwmple a
xi ss−t ,an
onalif f
t − e yo t e n
δ(rrα
)isnn
−
onze
s17
5]
r
dtipro
rat
−
it
teis
@
a
o
al . T hu o fin
y in
a
s − parenlef t e
p
l e−te lyd
S − threeα) ≤W isD( semicolon − Qm
⇔ DP),l n
a
∈
N|
←
← ∃arrowsouthwest − nt n − negationslash − arrowlef tuT { s ← ∀ n . ∀wnegationslash−arrowle
Lemma
1 1 . F or
an y s − et C, f − i C ≤ W (DX ; D¬X), the n
C ×C ≤ W
DX
(D XD ¬X ).A
P r o o . e f be on nu san dw i es s C≤ W
s su m
C ⊆ Y ( so m top o o i c a pa e N.T h en 0f(y ))eξ0 s fo r al Y.A
n
i 0 Teo r e 3 , w e p ce β ∈ N w th 0β, w here β (n = β(n)+ 2
embr s − h p i n C3 W e sh w C3 × C ≤ W
DX b y de finin g φβy)t
o eth
h
u t f ayi g f( ) (wose0d
ns t y a lways x t) on A0 h e ens,n
α o n A1, he od d s, whe r e α c o esf o m t h e c ano n ic l c o
ns u t n wt i npu {xnn∈ N w h ere
Am∩ ⇒, n) ] T 6=
x=
∅
J}∝F
f Ois
(1− 1/ prime−parenlef t
∃
parenright−parenright
delta−parenlef tf
(y
n)+
1/β( nparenr
imx
= δ(f (
∈ C3,
Hen ce , whe n β
1
(φ
(β,y)) =
1
2δparenlef t − f y))
xnot
xs ∈N d i v e s g e β e t m ≥ 1
{
eb
+ 2δf
n e g e r − period I n
a f i n it e
e
s p ac
h
NN, c n d − i e
e
A0
N
×N ×
A0 = C3
Aperiod − period1 = C3
× C3
NN
× NN ×
A..
Am−
× ... ×
NN × ..
.
NN,
×
N,
i+1
N
.
Am − 1
e m( Π3 )
and
Dm =
= C − three
¬
× three − C × .. × C3.
D∈
˜
wide
D
˜
Dwidem
= {h ii − lessm
∈ (NN mβ0 6∈
C3r − o
m − ax {
en
ow
sh o
wha
<
m
f − o r a n y Dm(Π3) s
0
˜
m is wideDmΠ3
) c o − m p l e t e .G i e n su ch a B
−
0⊇ B1 ⊇ ... B m−zero 1, Π3 s b e so
.
Def ie e f : 2N
f (α
)=
→ (N
0( α, f 1
N)m
α,
(
S
n − cth
.
)
mf
(
2→
y t n fi
).
1(α
)
− ).
e Bi s a ede re i − sn gi t i ss r a gh fo rw r − d
Bi⇔ f () ∈ C3
⇔
f( alpha − parenright
t
× ˜
Dm. Noti c eth a C3 widem=
Dm + 1.
∈
Ai
wh i c
t
c c kt h at α
o h
hsh o ws B ≤
∈
W
a d hen ot e a b o v e
i − nt−Ry−a m n d two − bracketright m n o − it e e r − li r , hr
r
w yud−l hb−e
Dm (Π 0)e − sS s uc
a
th at
DX ⊆ S
a
d S ∩ arrowdblright − logicalnotXequal − TJ .F einf inity − ut← existential −
e − hn .D−existential X − arrowlef t − negationslashT − equal{← Sintersection −
existential. arrowdblright − primeuniversal − unionmulti − commaarrowlef t −
a − T proportional − arrowlef t − prime − D − Jm 3inf inity − P i − D parenright −
comma−t−e−existential
arrowdbllef t − zero − three − owner − union← s − D.
∃arrowlef t −
negationslash
ich i − s c o n
W e s h l − ln owba i − s a l ysh owt hat X ≤ W
D X. L i − t er a l yth
s − ic − an n obt tu e be c − a u e X liv sina con nec eds p ce n − a d DX l
vesin ze r − o−dimen ion
sp ac e − period Howev r , r − o large e oug h ξ and alpha − comma int
r e t − c nga Dξ(Π0) set X
w i − t h Pth e r a − r t o a si n , 0], d − oes not hn ge h − t e Bre c as .S−i n c e P
hom e o morphict o N − N, f X ⊆ P oe−n c a nsho w
∗ r a li s wel n wo n (s e e5, p p . – 7o ex m pe ). G v en β ∈ NN,e
β
(n=
β(n +
N, let
1.F oreachn ∈
(13) n(β) = rn
=
1
∗
1
β(0 +
1
β(1) +
1
r
so
s
arrowdblright − onebracketright−unionmulti .T − T J ∝ hFphi−triangleinv ← (na)wasy,tenca
ab−so
r
l;φ (¬
n
N
N
oof Giv e β ∈ N ,l e f (β ∈2 b e the resu lto fr un ni ng thecn o n
)
whe xn i s the rn (β) i n( 13(
clc o nsr uci non inp u t {xn} n ∈N
,
w ih o nl yd e p end s o n β n +1) T Nhen δ( α) = l m nNn = φ(β). H e c
ef i
e snc
Pr
≤ (D
X
weae
one .
eo
h r e
f orTα≥
˜
2; wide
Dξ
)
m1
a
d
r
0
0
t
;D
),
0
e Γ
Πf ive − periodL
α f r beon α ≥
t he l
semicolon − threeeof
or Dξ c0ases Π
αo rΣα
orα ≥
3;
Dξ(Πα
ca, α≥
ω
adX
is
P o o f Th l h − en
u−n
Γ
s l o edX
d e
Γ isc o s e d u d ri nte s c − e tio n s
˜
w th Π02 se sal d Γwide
a − existential.dt∃ arrowlef t − negationslash − hs{Σ ← 0.S
h
←A s .∃
D
T
←∃
∞ ←←h
T m,
∩
φP
)]
∝F
←B
∃
T {←
.
T
∃←
←] ¬Xarrowlef t − negationslashO ←← ∃arrowlef t − mapstoarrowlef t − negationslash
W
3
X
t
o te poj eciv hea rc 0 c a ss s the n Xi
Γ cm pl
ttX Σ0 c om lp e − t),h−t en wDX is w Π
com pe t eparenright−period W en e − ed hec omp e − te s e comma − s{ Hn ⊆
n
obme a i cpr o p r es oft e i fu cto
% F rn a nd m ∈ N, le
N ot ce t
fΓ is
X
ha
hn , m = 12n + m
Thus h, : N
×
N → N is
on
parenright − parenlef tn + m
o n e - t on ean d ont .
+) +
e % : 2 N → 2N b
De n
i %(αparenright − parenlef tn) = o1 −⇔ ∀m
m.
,no mif i=0)
(α(
Aga n , if wethin kof α asa n N × N m t r ixo z ros n d ne s , w
mn m be ni g α(nmiparenright − commah − t n
(() = 0 i fro w n o f αco nains a 1 ;
1 i r o n o f α i i d e t i T hu i α it h b n ry se u n ≤ N h eer α n m)
α(hnmki ,h e %∗ α)n<N
w her e . O ne cane e n d%t o 2 b y e f ini
Dn − m
( )
t
= {nN
segmeo
s(n)
=1
⇔ f ra lhn
∈ N∃k
∀≥m
=
| hn, 0i < k}
f N)a
n∈D
ndf or
∗
The
y
mi ∈ D
o
,
)
m s), s(nm
i)
=0
k((αm ) n = %(
( i i − parenright Let H1 = {0} ⊆ 2N a nd Hn + 1 = % ←( Hn). Th e n n − H i − sΠ zero − n−
a y s t th f o − r ach i ∈ N, th a p r − poxima t o n − sαasteriskmath − n() = %
ev ntu al yqe u a t o %(α )i.
L e m m a 16 F o r H ⊆ 2 a d
X ⊆ [0 1, f
H ≤W
X
X
D ; D¬ X) n pa r i ua , f o r
≥ 2 a dX ⊆ [1 , Hn≤ W
X , th 1 X W
1 X
t i u o u , xn
t hatα m
g
s e ef (α b
a n d{ α} n ∈ N
,
= n)
)( an d
n∈ →%← ∞(xnH) =⇔ (%α
(α) ∈ H δ(f α)
⇔
g(α)) = lim
thecan
conv ege sp
lw
= (f (α)) ∈
x
s
ayex
st s .H
X⇔
(%
So%
o nt wise to% α,w s
f (α)
n ce−comma
∈D
n→∞ n
← (H) ≤W (DX; D
δ
.
X
X).
¬
0]is
Dξ(Π)−
cα o
D e re m Π0− comp e e
mp et e f o r α. ≥ 2 h n D
Xi − s
DΠ
˜
wide
com pe e
e − parenlef t Πzero − one + αhyphen − cmo ple t e
e − n DX is Dxi
˜
v ) f X ⊆ [0, 1] s Dwide m(Π0 )− c m pet efo r
1
mp l e − t e . 2 m+ Π
tioL n sr v rs el f o r
“ c m pl e e a nd a l i m
sαh≥3.w i
h
“ hard00 r ep l c g
c
e
e
P o o − f T he u pp r − eb oun ds for DX a re r m Prop si no 6an dCo
ro la r 7 . T
g
ic 1
l denot eb e y nt. N e e n w , i )H s j − ue wstt e − h e
hwe w l
h
second pa r w tf w e t h αmma< 16 F woth
e re m a nin c a − s e
Dξ = {
β α
0 n
∈H
a t
nd
h
hαiβ < ξ ∈
he l east β s u cht a t α
β n
Dξ
=¬ Dξ=
2N)ξ |
element − negationslashH i s o dd } ,
{h ββ <ξ ∈ (
)ξ
the leas
h − er e (Hn) isi c − n u d e d i
˜
e
is D(Π0)hyphen − c omp lete , n − a d Dξn
is wideD(Πhyphen
− parenrightc om p l − e te
m
er
eby a p p l − y i n g % c ord i n a
o
iβ < ξ ∈ Dξ Thu , we s − im p ym im c − i the p
i onth ehyp1 6th Lsis w Γ he aeX iso tΓ− c cm p
lee
r Γ − nharod D Le
ΠΓ nmbparenright−comma
thec a
e , r t o . t e
ξ
X. S Nnξ ξ
t − hI se u − tnen
ece
s c ou t abe , (2)
s ho meo
→c
th,α n =in {Nisin
α beta−parenlef tk) e|
n
af in i ts et co tai
in an i n ia segm e ∗ of e ac h β W ∗
e ca n t ea pl t ∗aech n tia se g me nt, bt a i n ng ayh αbeta −
commanβ < ξ. Le ~α be ∗ hex tns iono
hαβn iβ < by se ting a l un de f in d a l Nξ t z r T h∗ {~α}n∈ Nc o ver g
p o ntw s e t − oh(β iβ < ξ G ve n~ ∈ (2 ), el t x n = g(~n ) ∈[0 1]
w h r e~ i sa s ab o v e .T hne xn d pe ndso nl y o n af i n e p i e c of α
~.
f w es et f ~) t ob h e re ul t ofr u nn ing he ca n o n ca lcons r u tio n n
x − n} n ∈ N h − ten f is c n
asi n
Le
an
uΓ−ha
tin∗oiss dd.T h uswe a 6
fr,ex
mma
(D
edone
1
wh ic h f l − lo w s i m
R eferen cs [ ]
L.K
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