Barely lonely runners and very lonely runners: a refined approach to the Lonely Runner Problem

Abstract

We introduce a sharpened version of the well-known Lonely Runner Conjecture of Wills and Cusick. Given a real number x, let x denote the distance from x to the nearest integer. For each set of positive integer speeds v 1 , . . . , v n , we define the associated maximum loneliness to beThe Lonely Runner Conjecture asserts that ML(v 1 , . . . , v n ) 1/(n + 1) for all choices of v 1 , . . . , v n . We make the stronger conjecture that for each choice of v 1 , . . . , v n , we have either ML(v 1 , . . . , v n ) = s/(ns + 1) for some s ∈ N or ML(v 1 , . . . , v n ) 1/n. This view reflects a surprising underlying rigidity of the Lonely Runner Problem. Our main results are: confirming our stronger conjecture for n 3; and confirming it for n = 4 and n = 6 in the case where one speed is much faster than the rest.