Etude des sous-groupes de (R,+)

◦
(R,+)
G(R,+) G∗
+=G∩R∗ +
G∗
+
G∗
+a a >0
a > 0
x G ]a; 2a[ (1)
x0G[a;x[
x−x0
a∈G
aZ={an / n ∈Z}G
aZ=G
t∈G∗
+n=t
a
t−na t =na
a= 0 GR
GR∀(x, y)∈R2, x < y ⇒ ∃t∈G x < t < y
λ µ λ < µ g G 0< g < µ −λ
n n > λ
g
n0
n0g∈]λ;µ[
(R,+)
αZ[α] = {a+αb / a ∈Z, b ∈Z}
Z[α] (R,+)
αZ[α] = x0Zx0
α
Z[α]α
(O,~ i,~ j) (Γ) Mθ
(Γ) (
~
i, ~ OM)θ
EZ={Mn/ n ∈Z}(Γ)
∀(θ, θ0)∈R2, θ < θ0⇒ ∃n∈Z/(
~
i, ~ OMn)θ θ0.
Z[2π]
F={cos(n)/n ∈N}[−1; 1]
HR∗
+
•α > 0H={αn/n ∈Z}
•HR∗
+