Given $n$, what is the smallest value $\delta_n$ satisfying the following:
For any group of $n$ runners with constant but distinct speeds,
starting from the same point and running clockwise along the unit
circle, there exists a moment, at which the length of each of $n$
empty circle segments between two consecutive runners does not exceed
$\delta_n$.
More formally, for $n \in \mathbb{N}$, $M=\{m_1,\dots,m_n\} \subset \mathbb{N}$, and $t \in [0,1]$, we sort the fractional parts of the values $tm_i$ in ascending order: $0 \le \{tm_{i_1}\} \le\dots\le \{tm_{i_n}\} \le 1$. Then we consider the longest empty segment between two consecutive 'runners':
$$ \text{LongestArc}(M,t) := \max\Big\{ \{tm_{i_2}\}-\{tm_{i_1}\}, \dots, \{tm_{i_n}\}-\{tm_{i_{n-1}}\}, 1+\{tm_{i_1}\}-\{tm_{i_n}\} \Big\}.$$
Finally, we put
$$ \delta_n := \mathop{\smash{\mathrm{sup}}}_{|M|=n} \inf_{t \in [0,1]} \text{LongestArc}(M,t).$$
Has the problem of finding these values (or, perhaps, an equivalent one) been studied earlier? I have looked through some papers related to the Lonely Runner Conjecture (including this one) but found nothing.
It is clear that $\delta_2=\frac12$. With a little more effort goes $\delta_3=\frac25$, one of the 'extremal' sets here is $M=\{0,1,3\}$. One may check that $\delta_4 \ge \frac13$ by taking $M=\{0,1,2,4\}$. I suspect this bound to be tight. Maybe someone sees a counterexample or a short argument, why is that so?
For large $n$, I believe I can show that $\delta_n \ge \frac{\log n -O(\log \log n)}{n}$. However, I don't have any (decent) upper bound on $\delta_n$ apart from the fact that $\delta_n=o(1)$ as $n \to \infty$.
Finally, I found that the 'dual' problem concerning the values
$$ \epsilon_n := \inf_{|M|=n} \mathop{\smash{\mathrm{sup}}}_{t \in [0,1]} \text{LongestArc}(M,t)$$
has earlier been considered on MathOverflow (not in the literature, though). One may also state two more 'dual' extremal problems related to the shortest arc (instead of the longest one), but I haven't found anything published about them as well.
I would appreciate any related reference!
Best Answer
I think the answer to the question as posed is "yes", see Konyagin-Ruzsa-Schlag (2011)
https://gauss.math.yale.edu/~ws442/papers/DILATES.pdf
It looks like nothing beyond the bounds discovered in the comments and Fedja's answer above is known.