Dirichlet Approximation Theorem
\forall \alpha \in \mathbb{R}:\, \lfloor \alpha \rfloor := \max \{n \in \mathbb{Z} \,|\, n \leq \alpha\} \in \mathbb{Z}
\forall \alpha \in \mathbb{R}:\, \{\alpha\} := \alpha - \lfloor\alpha\rfloor \in [0, 1)
\forall n \in \mathbb{N}_0:\, [n]_0 := \{k \in \mathbb{N}_0 \,|\, 0 \leq k \leq n \}
Definitions and notation
Dirichlet Approximation Theorem
\forall\, \alpha, Q \in \mathbb{R}, Q \geq 1\,\exists\, p, q \in \mathbb{Z},\, 1 \leq q \leq Q:
\left|q\alpha - p\right| \leq \frac{1}{\lfloor Q \rfloor + 1} < \frac{1}{Q} \leq \frac{1}{q}
PROOF:
\bar{Q} := \lfloor Q \rfloor + 1 \in \mathbb{N}
A:=\left\{\{k\alpha\right\}\,|\,k \in [\bar{Q}]_0\},\,|A| = \bar{Q} + 1
\forall\, \alpha, Q \in \mathbb{R}, Q \geq 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:
\left|\left\{\frac{p}{q} \in \mathbb{Q}:\,\left|\alpha - \frac{p}{q} \right| < \frac{1}{q^2}\right\}\right| = \aleph_0
\forall \alpha \in \mathbb{R} \setminus \mathbb{Q}:
Dirichlet Approximation Theorem
PROOF:
\left|\left\{\frac{p}{q} \in \mathbb{Q}:\,\left|\alpha - \frac{p}{q} \right| < \frac{1}{q^2}\right\}\right| < \aleph_0
\forall \alpha \in \mathbb{Q}:
Dirichlet Approximation Theorem
PROOF: