Problems on track runners (Q2306363)

The paper under review is concerned with some track runner problems in the spirit of the famous Lonely Runner Conjecture. It is shown that given a circular arc \(A\) on a circular track of unit circumference, one can find a \(k \in \mathbb{N}\), \(k\) starting positions and \(k\) distinct, constant speeds, such that for \(k\) runners starting at these positions and running at these speeds, at least one runner will always be in the arc \(A\). The constructed schedule of runners in this case has each runner run at a rational speed. Conversely, it is deduced from Kronecker's Theorem that for \(k\) runners with rationally independent speeds, there exists arbitrarily large \(t\), such that at time \(t\), all runners are in \(A\). The same is shown to hold if one has the starting positions fixed, but is free to choose rational speeds of the \(k\) runners. The paper ends in some algorithmic aspects of the problem of deciding whether a schedule of \(k\) runners with the properties above (in the two cases) exists.