Francisco Santos: Lattice zonotopes and the lonely-runner conjecture

Abstract: The lonely runner conjecture (LRC) is the following statement formulated by Jörg Wills in 1968:

If $n$ ``runners’’ move along a circle of length one, all starting at the origin, each with its own (constant) velocity, then there is a time at which they are all at distance at least $1/(n+1)$ from the origin.

In its shifted version (sLRC) the runners are allowed to start each at a different position and the velocities are assumed distinct (or else giving all runners the same velocity produces an easy counter-example). The conjecture has been approached from different perspectives, and is proved until $n=6$ in the original version (Barajas and Serra 2008) and $n=3$ in the shifted version (Cslovjecsek et al. 2022). The latter is based in a reformulation of LRC and sLRC as questions about the covering radii of certain zonotopes, a connection developed by Malikiosis and Schymura (2017).

In this talk I will review the conjecture and its relation to zonotopes and covering radii, and will show that, both in the original and the shifted versions, if the conjecture holds for integer velocities adding up to at most $n^{2n}$ then it holds for arbitrary velocities. This improves a recent result of Terence Tao, who proved the same with a bound of $n^{Cn^2}$ instead. We then use this bound to computationally prove the shifted version of the conjecture in the first open case, $n=4$.

The first part is joint with Malikiosis and Shymura, and the second part with Alcántara and Criado.