This is just a completely random question of the top of my head.
Let $\Omega$ be the set of bijections $\phi:\mathbb{N} \cup \{0\} \rightarrow \mathbb{Q}$.
We can imagine any $\phi \in \Omega$ to be equivalent to a wacky "staircase", where for $x \in \mathbb{R}^+$ we have $x \mapsto \phi(\lfloor x \rfloor)$.
If we walk along this staircase, the biggest step we need to take is of size $\sup_{n \geq 0} |\phi(n+1)-\phi(n)|$. Suppose we want to make the steps as small as possible. How small can we make them? That is, what is \[\inf_{\phi \in \Omega} \sup_{n \geq 0} |\phi(n+1)-\phi(n)|\ \ ?\]