While reading several articles about lattice basis reduction I am left with a few questions.

For one, I came across this piece of text

Let $\alpha$ and $\beta \in \mathbb{R}$. Also let $X>0$ and $X$ is large. Then to compute $x,y \in \mathbb{Z}$ with $\text{max} (|x|,|y|) \le X$ and such that $|\alpha x + \beta y|$ is minimal we apply the lattice basis reduction to the lattice generated by the columns ${1 \choose C\alpha}$ and ${0 \choose C\beta}$ for $C$ large enough.

My question is where is the $C$ coming from? When is it large enough? It obviously depends on something... maybe on $X ?$

All hints, examples or explanations are very much welcome.