Approximations diophantiennes 1. Fractions continues

(an)
[a0, a1, . . . , an] = a0+1
a1+1
a2+1
+1
an
=cn
cn=pn
qn
pn=anpn−1+pn−2
qn=anqn−1+qn−2p−1= 1 p0=a0 q−1= 0 q0= 1
pn+1qn−pnqn+1 = (−1)npnqn−2−pn−2qn= (−1)nan
cn+1 −cn=(−1)n
qn+1qncn−cn−2=(−1)nan qn−2qn
pn
pn−1
= [an, an−1, . . . , a0]qn
qn−1
= [an, an−1, . . . , a1]
α[a0, a1, a2, . . .]
ai∈N∗i

(αi) ]0,1[
α=a0+1
a1+1
a2+1
+1
an+αn
=pn+pn−1αn
qn+qn−1αn
|qnα−pn|=βn=
n
Y
i=1
αn
ΦNP1
Φ(n)α
an≤Φ(n)P1
Φ(n)α
an≤Φ(n)
ΨN
PΨ(n)α|qα −p|<Ψ(q)
PΨ(n)α|qα −p|<Ψ(q)
α
ω(α) = sup ν, |qα −p|<1
qν(p, q)∈Z2.
µ(α) = ω+ 1
α θ (c, θ)c > 0
|qα −p|> c ·q−θp q
θ θ
ω(α)< θ ⇒α θ ⇒ω(α)≤θ
ω(α) = lim sup
n→+∞
ln qn+1
ln qn
= 1 + lim sup
n→+∞
ln an+1
ln qn
.

α
β(α) = sup ν, |qα −p|<1
νq(p, q)∈Z2.
β(α) = lim sup
n→+∞
ln qn+1
qn
= lim sup
n→+∞
ln an+1
qn
.
α β(α)=1
B(α) = X
n>0
βn−1ln(α−1
n).
α
X
n>0
ln qn+1
qn
<+∞
B(α)<+∞
B(α) = −ln α+α·B(1
α)
α∈]0,1[
α⇒α⇒β(α)=1