How to minimize expressions of the form $|Ax+By+C|$ with $x$ and $y$ integers and $A,B,C \in \mathbb R$? pretty straight forward question, I know how to do it if $A,B \in \mathbb Z$ (I believe it is the $\frac{C}{\gcd(A,B)}$).
But what if $A,B$ and $C$ are just given real numbers with no gcd?
 A: Good question! In order to say something meaningful, I'll consider the simplified problem where $C=0$. That is, given $A,B\in\mathbb{R}$, find integers $x,y$ with $|Ax+By|$ minimal. One problem: This is trivial as we can just take $x=y=0$. So let's impose an additional restriction: $y$ can't be zero. Moreover, if $A=0$ we end up with $|By|$ which is minimized by $y=1$. Not very interesting either, so we'll say that $A\neq 0$.
Now we can rewrite the expression we want to minimize as
$$|A|\cdot |y|\cdot\left|\frac xy+\frac BA\right|.$$
Here, the $|A|$ factor isn't very interesting (why?), and $B/A$ is just any real number. We now reformulate the question a little bit:
Reformulated question: Let $\xi\in\mathbb{R}$ be any real number. For what $(x,y)\in\mathbb{Z}^2$ with $y\neq 0$ is
$$
|y|
\cdot \left|\xi-\frac xy\right|
$$
minimal?
This question has an answer, and it is given by the following famous result.
Dirichlet's Approximation Theorem: Let $\xi\in \mathbb{R}\setminus\mathbb{Q}$ be an irrational real number, then there exist infinitely many pairs of integers $(x,y)\in\mathbb{Z}^2$ with $y\neq 0$, $\gcd(x,y)=1$ and
$$
\left|\xi-\frac xy\right|<\frac1{y^2}.
$$
Note that for any such pair, $|y|\cdot\left|\xi-\frac{x}{y}\right|<\frac1{|y|}$, which can be as small as we want it to be! Just pick a pair with $|y|$ large enough. Therefore, the infimum is zero. However, for irrational values of $\xi$, there is no pair $(x,y)$ that actually gives zero, so the minimum doesn't exist.

One can also ask if we can do better than Dirichlet's Approximation Theorem. As it turns out, if $\xi\in\mathbb{R}\setminus\mathbb{Q}$ is algebraic, we cannot:
Roth's Theorem: Let $\xi\in\mathbb{R}\setminus\mathbb{Q}$ be an irrational algebraic number, then for all $\varepsilon>0$, there are only finitely many pairs $(x,y)\in\mathbb{Z}^2$ with $y\neq 0$, $\gcd(x,y)=1$ and
$$
\left|\xi-\frac{x}{y}\right|<\frac1{q^{2+\varepsilon}}.
$$

How do you actually find the $(x,y)$ in Dirichlet's theorem? I don't know, but seeing as a common proof uses Minkowski's first convex body theorem, efficiently finding good $(x,y)$ probably involves use of the LLL-algorithm.
