The lonely runner conjecture states that for n runners on a unit-length track with constant, nonzero, integer speeds, all starting from the same position, there exists a time t when each runner is at least 1/(n+1) units away from the start line. This conjecture remains open for seven or more runners. For a given set of speeds, the maximum loneliness is defined to be the largest value L for which there is a time t at which every runner is at least L units away from the start line.
In this talk, I will introduce a related concept called the lonely runner spectra. Recent work by Noah Kravitz and Vikram Giri shows that these spectra possess a rich "hierarchical" structure. I will describe how these relative spectra exhibit rigid arithmetic properties and how each spectrum can be fully characterized by a finite computation. Finally, I will outline how such a computation can be used to completely characterize the maximum loneliness values for three runners up to any value strictly greater than zero. Based on joint work with Noah Kravitz.
