Let $x\_1,...,x\_N$ be $N$ mutually distinct real numbers.


I wonder: How does the expression


$$ f\_x(N)=\sup\_{\lambda \in \mathbb R}\min\_{i \neq j} \min\_{n \in \mathbb Z}\left\vert \lambda (x\_i-x\_j)-n\right\rvert$$
depend on $N$? I acknowledge the answer might depend on $x$, but I am looking for a lower bound on $\inf\_{x; all x\_i distinct}f\_x(N)$


Let's say we have two numbers, then we can choose $\lambda (x\_1-x\_2)=1/2.$ Hence, the optimal value in that case is $1/2.$



Maximal distance of set to integers
If I understood correctly, your question is a form of the [Lonely runner conjecture](https://en.wikipedia.org/wiki/Lonely_runner_conjecture). You can see several results and a summary on that Wikipedia link. An important special case is when the difference of the $x\_i$'s are irrational, in which case you can spread them so that you can get arbitrarily close to $1/N$.


