Dirichlet Approximation Theorem
∀
α
∈
R
:
⌊
α
⌋
:
=
max
{
n
∈
Z
∣
n
≤
α
}
∈
Z
\forall \alpha \in \mathbb{R}:\, \lfloor \alpha \rfloor := \max \{n \in \mathbb{Z} \,|\, n \leq \alpha\} \in \mathbb{Z}
∀
α
∈
R
:
⌊
α
⌋
:=
max
{
n
∈
Z
∣
n
≤
α
}
∈
Z
\forall \alpha \in \mathbb{R}:\, \lfloor \alpha \rfloor := \max \{n \in \mathbb{Z} \,|\, n \leq \alpha\} \in \mathbb{Z}
∀
α
∈
R
:
{
α
}
:
=
α
−
⌊
α
⌋
∈
[
0
,
1
)
\forall \alpha \in \mathbb{R}:\, \{\alpha\} := \alpha - \lfloor\alpha\rfloor \in [0, 1)
∀
α
∈
R
:
{
α
}
:=
α
−
⌊
α
⌋
∈
[
0
,
1
)
\forall \alpha \in \mathbb{R}:\, \{\alpha\} := \alpha - \lfloor\alpha\rfloor \in [0, 1)
∀
n
∈
N
0
:
[
n
]
0
:
=
{
k
∈
N
0
∣
0
≤
k
≤
n
}
\forall n \in \mathbb{N}_0:\, [n]_0 := \{k \in \mathbb{N}_0 \,|\, 0 \leq k \leq n \}
∀
n
∈
N
0
:
[
n
]
0
:=
{
k
∈
N
0
∣
0
≤
k
≤
n
}
\forall n \in \mathbb{N}_0:\, [n]_0 := \{k \in \mathbb{N}_0 \,|\, 0 \leq k \leq n \}
Definitions and notation
Dirichlet Approximation Theorem
∀
α
,
Q
∈
R
,
Q
≥
1
∃
p
,
q
∈
Z
,
1
≤
q
≤
Q
:
\forall\, \alpha, Q \in \mathbb{R}, Q \geq 1\,\exists\, p, q \in \mathbb{Z},\, 1 \leq q \leq Q:
∀
α
,
Q
∈
R
,
Q
≥
1
∃
p
,
q
∈
Z
,
1
≤
q
≤
Q
:
\forall\, \alpha, Q \in \mathbb{R}, Q \geq 1\,\exists\, p, q \in \mathbb{Z},\, 1 \leq q \leq Q:
∣
q
α
−
p
∣
≤
1
⌊
Q
⌋
+
1
<
1
Q
≤
1
q
\left|q\alpha - p\right| \leq \frac{1}{\lfloor Q \rfloor + 1} < \frac{1}{Q} \leq \frac{1}{q}
∣
q
α
−
p
∣
≤
⌊
Q
⌋
+
1
1
<
Q
1
≤
q
1
\left|q\alpha - p\right| \leq \frac{1}{\lfloor Q \rfloor + 1} < \frac{1}{Q} \leq \frac{1}{q}
PROOF:
Q
ˉ
:
=
⌊
Q
⌋
+
1
∈
N
\bar{Q} := \lfloor Q \rfloor + 1 \in \mathbb{N}
Q
ˉ
:=
⌊
Q
⌋
+
1
∈
N
\bar{Q} := \lfloor Q \rfloor + 1 \in \mathbb{N}
A
:
=
{
{
k
α
}
∣
k
∈
[
Q
ˉ
]
0
}
,
∣
A
∣
=
Q
ˉ
+
1
A:=\left\{\{k\alpha\right\}\,|\,k \in [\bar{Q}]_0\},\,|A| = \bar{Q} + 1
A
:=
{
{
k
α
}
∣
k
∈
[
Q
ˉ
]
0
}
,
∣
A
∣
=
Q
ˉ
+
1
A:=\left\{\{k\alpha\right\}\,|\,k \in [\bar{Q}]_0\},\,|A| = \bar{Q} + 1
∀
α
,
Q
∈
R
,
Q
≥
1
:
\forall\, \alpha, Q \in \mathbb{R}, Q \geq 1:
∀
α
,
Q
∈
R
,
Q
≥
1
:
\forall\, \alpha, Q \in \mathbb{R}, Q \geq 1:
∀
α
∈
R
∃
p
,
q
∈
Z
:
∣
α
−
p
q
∣
<
1
q
2
\forall\, \alpha \in \mathbb{R}\,\exists\, p, q \in \mathbb{Z}:\,\left|\alpha - \frac{p}{q}\right| < \frac{1}{q^2}
∀
α
∈
R
∃
p
,
q
∈
Z
:
α
−
q
p
<
q
2
1
\forall\, \alpha \in \mathbb{R}\,\exists\, p, q \in \mathbb{Z}:\,\left|\alpha - \frac{p}{q}\right| < \frac{1}{q^2}
Dirichlet Approximation Theorem
PROOF:
∣
{
p
q
∈
Q
:
∣
α
−
p
q
∣
<
1
q
2
}
∣
=
ℵ
0
\left|\left\{\frac{p}{q} \in \mathbb{Q}:\,\left|\alpha - \frac{p}{q} \right| < \frac{1}{q^2}\right\}\right| = \aleph_0
{
q
p
∈
Q
:
α
−
q
p
<
q
2
1
}
=
ℵ
0
\left|\left\{\frac{p}{q} \in \mathbb{Q}:\,\left|\alpha - \frac{p}{q} \right| < \frac{1}{q^2}\right\}\right| = \aleph_0
∀
α
∈
R
∖
Q
:
\forall \alpha \in \mathbb{R} \setminus \mathbb{Q}:
∀
α
∈
R
∖
Q
:
\forall \alpha \in \mathbb{R} \setminus \mathbb{Q}:
Dirichlet Approximation Theorem
PROOF:
∣
{
p
q
∈
Q
:
∣
α
−
p
q
∣
<
1
q
2
}
∣
<
ℵ
0
\left|\left\{\frac{p}{q} \in \mathbb{Q}:\,\left|\alpha - \frac{p}{q} \right| < \frac{1}{q^2}\right\}\right| < \aleph_0
{
q
p
∈
Q
:
α
−
q
p
<
q
2
1
}
<
ℵ
0
\left|\left\{\frac{p}{q} \in \mathbb{Q}:\,\left|\alpha - \frac{p}{q} \right| < \frac{1}{q^2}\right\}\right| < \aleph_0
∀
α
∈
Q
:
\forall \alpha \in \mathbb{Q}:
∀
α
∈
Q
:
\forall \alpha \in \mathbb{Q}:
Dirichlet Approximation Theorem
PROOF:
1
.
1
Resume presentation
Dirichlet Approximation Theorem
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