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](https://en.wikipedia.org/wiki/Lonely_runner_conjecture) (including [this one](https://arxiv.org/abs/1508.07289)) 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](https://mathoverflow.net/questions/176822/not-lonely-runners) 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!



Has the following problem, resembling the lonely runner conjecture, been studied?
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](https://gauss.math.yale.edu/%7Ews442/papers/DILATES.pdf)


It looks like nothing beyond the bounds discovered in the comments and Fedja's answer above is known.


