Transcendence criteria for pairs of continued fractions
9
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Découvre YouScribe et accède à tout notre catalogue !
Découvre YouScribe et accède à tout notre catalogue !
9
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
- let
- nite words
- qsn?1 qsn
- trans enden
- dire tion
- pn?1
- either trans
- positive integers
Publié par
Langue
English
′α (α,α )
α
α
α
′(α,α )
(α,α)
if
Ad
one
giving
and
illustrate
Y
of
ann
um
Bugea
ds
ud
um
Univ
b
ersit?
Claude
,
Bernard
one
Ly
on
on
ued
1
partial
and
Univ
ts
ersit?
either
Louis
t
P
t
asteur,
a
F
rance
pro
Abstra
if
e
The
Subje
aim
ued
of
2
the
if
presen
a
t
has
note
y
is
ks
to
usually
establish
m
t
w
note
o
of
extensions
a
of
of
some
ensuring
transcendence
um
e
for
V
real
w
n
some
um
transcenden
b
in
ers
giv
TI?KI
en
b
11J81,
y
T
their
[1
tin
3].
ued
fraction
expansions.
ts
W
irrational
e
er
adopt
sp
the
prop
follo
for
wing
blo
p
partial
oin
eat
t
to
of
then
view:
b
rather
tal,
than
purp
giving
presen
sucien
to
t
o
of
ensuring
e
the
tly
transcendence
oin
of
rather
a
t
giv
transcendence
en
en
n
er
um
e
b
pair
er
41(61)(2006),
Boris
real
,
ers,
w
aim
e
that,
tak
at
e
them
a
Clearly
pair
kno
CTIONS
FRA
them
CONTINUED
if
OF
the
AIRS
MA
P
Sc
of
2000
real
Classic
n
Key
um
phr
b
ers,
223
and
tin
w
fractions
e
,
pro
,
v
They
e
the
that,
that
under
the
some
of
quotien
at
of
least
real
one
n
of
b
them
F
is
some
transcenden
ecial
tal.
binatorial
1.
ert
Intr
,
oduction
example
and
long
resul
ts
of
V
quotien
ery
rep
little
un
is
kno
the
wn
eginning,
on
CRITERIA
the
ust
e
tin
transcenden
ued
or
fraction
The
expansion
ose
of
the
an
t
y
is
establish
real
w
n
extensions
um
some
b
our
er
W
of
adopt
degree
sligh
at
dieren
least
p
three.
t
It
view:
is
than
lik
sucien
ely
that
the
the
of
giv
of
n
its
b
partial
TRANSCENDENCE
quotien
w
ts
tak
is
a
un
231
b
223
ounded,
ol.
but
of
w
n
e
b
seem
and
to
e
b
at
e
ving
still
under
v
ery
least
far
of
a
is
w
tal.
a
,
y
one
from
ws
OR
adv
,
that
plainly
of
es
is
transcendence
or
Lik
w
in
2
pair
3
TEMA
the
GLASNIK
ofs
a
this
pro
giv
of.
a
tly
e
,
[1,
a
,
small
℄
step
pro
w
rest
as
the
made
hmidt
in
theorem.
this
Mathematics
b
ation.
y
11J70.
means
wor
of
and
sev
ases.
eral
new
tin
transcendence
fractions.
forA W A
W |W|
W := a ...a W := a ...a1 m m 1
W W =W
′ ′a = (a ) a = (a ) Aℓ ℓ≥1 ℓ≥1ℓ
′ ′a a ... a a ...1 2 1 2
′(a,a ) (∗)
(V )n n≥1
n≥ 1 V an
′n≥ 1 V an
(|V |)n n≥1
′a a
(∗)
′ ′ ′α = [0;a ,a ,...], α = [0;a ,a ,...].1 2 1 2
′α α
a
′a
(W )j j≥0
Z X = W X = X W X j ≥ 1≥1 0 0 j j−1 j j−1
(X ) (X )j j≥0 j j≥0
′ ′a = (a ) a = (a )ℓ ℓ≥1 ℓ≥1ℓ
′ ′ ′α = [0;a ,a ,...], α = [0;a ,a ,...].1 2 1 2
′α α
′a = a
a = (a )ℓ ℓ≥1
a α :=
[0;a ,a ,...,a ,...]1 2 ℓ
′(a,a ) (∗∗)
(U ) (V )n n≥1 n n≥1
n≥ 1 V an
′n≥ 1 U V an n
,
of
Condition
es
the
the
se
se
e
t
b
h
and
the
Then,
1.3
either
in
one
(at
e
le
to
ast)
a
of
ansc
et
Applying
and
1],
L
osing
.
If
is
the
tr
ord
ansc
atic
endental,
tin
or
pair
b
t
oth
results,
ar
AMCZEWSKI
e
(ii)
in
or
the
oth
same
al
r
mirror
e
v
al
w.
quadr
b
atic
er
eld.
p
W
b
e
p
stress
numb
that
et
there
length
is
Let
no
endental.
assumption
wider
on
the
the
sa
gro
wth
use
of
of
the
T
1.1
ev
and
ord
Theorem
w
satisfying
.
w
W
is
e
or
p
e
oin
r
t
atic
out
1.1
t
.
w
e
o
[3,
is
b
;
ollar
of
L
Theorem
is
1.1.
letters
Cor
e
ollar
e
y
inte
1.2
wor
.
e
L
arily
et
omes,
e
the
um
(iii)
that
;
the
ord
a
w
set.
the
e
b
either
e
tr
an
next
arbitr
with
ary
of
se
fractions.
o
e
W
of
that
nite
w
wor
ds
terminology
on
there
the
o
alphab
w
et
enien
of
it
prex
state
a
BUGEA
.
(i)
Set
Y.
is
the
ord
B.
w
of
the
224
,
ev
and
.
ery
Set
ev
inte
for
image
(ii)
tr
;
endental,
ord
b
w
ar
the
in
of
same
prex
e
a
quadr
is
eld.
ord
Theorem
for
with
any
The
w
y
the
w
,
.
er
Then,
Theorem
the
stated
se
elo
Cor
es
y
ery
.
ev
et
for
denoted
(i)
,
that:
h
of
b
and
a
ords
w
of
nite
ositive
of
gers.
the
a
d
exists
b
gins
onver
arbitr
ge.
long
Denote
alindr
their
then
limits
r
by
al
there
er
if
n
Condition
is,
satises
,
pair
alphab
the
on
that
w
y
of
sa
The
and
table
e
a
W
b
.
is
ely
quadr
ectiv
or
resp
ansc
,
Our
and
statemen
ords
deals
w
a
innite
,
r
ued
esp
Keep
e
ab
v
and
notation.
set
e
the
y
with
the
tify
ords.
iden
on
e
bi-
w
satises
that
from
,
the
from
if
ts
exist
elemen
w
of
nite
e
ords
b
t
and
v
Let
is
.
and
if
our
only
o
and
UD
if
palindrome
that:
a
for
is
ery
particular,
AND
In
,
.
w
ord
AD
w
is
the
prex
is
the
of
ord
ersal
;
rev
for
Then
ery
at
Condition
gers
,
p
w
ord
ositive
of
a
le
is
ast
prex
one
the
among
ord
the
and
immediate
(|U |/|V |)n n n≥1
(|V |)n n≥1
′a a
(∗∗)
′ ′ ′α = [0;a ,a ,...], α = [0;a ,a ,...].1 2 1 2
′(p /q ) αℓ ℓ ℓ≥1
1/ℓ ′(q ) α αℓ≥1ℓ
′a =a
a
(a )ℓ ℓ≥1
b ,...,b (n )1 m k k≥1
(λ )k k≥1
a =b 1≤j≤m 0≤h≤λ −1,n +j+hm j kk
n >n +λ m k≥ 1k+1 k k
λk
limsup > 0,
nk→+∞ k
[0;a ,a ,...]1 2
a
b ,...,bm 1
m≥ 2 L ,.
