Simultaneous Diophantine approximation

Question

Is the following statement true? For all $\alpha,\beta\in \mathbb R$ and for all $\varepsilon \in \mathbb R_{>0}$, there exist $a,b,c\in \mathbb Z$ such that $|a-c\alpha|<\varepsilon$ and $|b-c\beta|<\varepsilon$?

Answer 1

The answer to this question lies in the simultaneous version of the Dirichlet's Approximation Theorem. A sketch of the proof using the pigeonhole principle is presented in the link.

In the case with only two reals ($\alpha$ and $\beta$), it states that for any integer $N$, we can find $p_{1} , p_{2} \in \mathbb{Z} $, and $1 \leq q \leq N$ such that

$$ \begin{cases} \left| \alpha - \frac{p_{1}}{q} \right| \leq \frac{1}{q N^{\frac{1}{2}}} \\ \left| \beta - \frac{p_{2}}{q} \right| \leq \frac{1}{q N^{\frac{1}{2}}} \end{cases} $$

Multiplying both sides by $q$ and taking $N \geq \frac{1}{\epsilon^{2}}$, your statement is proven.