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 $\beta \in \mathbb{R}$. Then there are two almost the same ways to compute small values for $\alpha x + \beta y$ with not too large $x,y \in \mathbb{Z}$.
1) applying the continued fraction algorithm
2) Applying the lattice basis reduction algorithm to the lattice generated by the columns of the matrix
\begin{pmatrix} 1 & 0 \\ C\alpha & C\beta \end{pmatrix} for $C$ large enough.
Why are those (for me different algorithms) in the above sense the same?
And also, where is the $C$ coming from? When is it large enough? It obviously depends on something...
All hints, examples or explanations are very much welcome.