Random runners are very lonely

Suppose that $k$ runners having different constant speeds run laps on a circular track of unit length. The Lonely Runner Conjecture states that, sooner or later, any given runner will be at distance at least $1/k$ from all the other runners. We prove that, with probability tending to one, a much stronger statement holds for random sets in which the bound $1/k$ is replaced by \thinspace $1/2-\varepsilon$. The proof uses Fourier analytic methods. We also point out some consequences of our result for colouring of random integer distance graphs

Correlation among runners and some results on the Lonely Runner Conjecture

The Lonely Runner Conjecture was posed independently by Wills and Cusick and has many applications in different mathematical fields, such as diophantine approximation. This well-known conjecture states that for any set of runners running along the unit circle with constant different speeds and starting at the same point, there is a moment where all of them are far enough from the origin. We study the correlation among the time that runners spend close to the origin. By means of these correlations, we improve a result of Chen on the gap of loneliness and we extend an invisible runner result of Czerwinski and Grytczuk. In the last part, we introduce dynamic interval graphs to deal with a weak version of the conjecture thus providing some new results.Comment: 18 page

Lonely runners in function fields

The lonely runner conjecture, now over fifty years old, concerns the following problem. On a unit length circular track, consider $m$ runners starting at the same time and place, each runner having a different constant speed. The conjecture asserts that each runner is lonely at some point in time, meaning distance at least $1/m$ from the others. We formulate a function field analogue, and give a positive answer in some cases in the new setting

Computing the covering radius of a polytope with an application to lonely runners

We are concerned with the computational problem of determining the covering radius of a rational polytope. This parameter is defined as the minimal dilation factor that is needed for the lattice translates of the correspondingly dilated polytope to cover the whole space. As our main result, we describe a new algorithm for this problem, which is simpler, more efficient and easier to implement than the only prior algorithm of Kannan (1992). Motivated by a variant of the famous Lonely Runner Conjecture, we use its geometric interpretation in terms of covering radii of zonotopes, and apply our algorithm to prove the first open case of three runners with individual starting points.Comment: 22 pages, 4 tables, 2 figures, revised versio