Some remarks on the lonely runner conjecture

The lonely runner conjecture of Wills and Cusick, in its most popular formulation, asserts that if $n$ runners with distinct constant speeds run around a unit circle $\R/\Z$ starting at a common time and place, then each runner will at some time be separated by a distance of at least 1/(n+1) from the others.  In this paper we make some remarks on this conjecture.  Firstly, we can improve the trivial lower bound of 1/(2n) slightly for large n, to (1/(2n)) + (c \log n)/(n^2 (\log\log n)^2) for some absolute constant c>0; previous improvements were roughly of the form (1/(2n)) + c/n^2.  Secondly, we show that to verify the conjecture, it suffices to do so under the assumption that the speeds are integers of size n^{O(n^2)}.  We also obtain some results in the case when all the velocities are integers of size O(n)

Some remarks on the lonely runner conjecture

The lonely runner conjecture of Wills and Cusick, in its most popular formulation, asserts that if $n$ runners with distinct constant speeds run around a unit circle $\R/\Z$ starting at a common time and place, then each runner will at some time be separated by a distance of at least 1/(n+1) from the others.  In this paper we make some remarks on this conjecture.  Firstly, we can improve the trivial lower bound of 1/(2n) slightly for large n, to (1/(2n)) + (c \log n)/(n^2 (\log\log n)^2) for some absolute constant c>0; previous improvements were roughly of the form (1/(2n)) + c/n^2.  Secondly, we show that to verify the conjecture, it suffices to do so under the assumption that the speeds are integers of size n^{O(n^2)}.  We also obtain some results in the case when all the velocities are integers of size O(n)

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

A few more Lonely Runners

Lonely Runner Conjecture, proposed by J\"{o}rg M. Wills and so nomenclatured by Luis Goddyn, has been an object of interest since it was first conceived in 1967 : Given positive integers $k$ and $n_1,n_2,\ldots,n_k$ there exists a positive real number $t$ such that the distance of $t\cdot n_j$ to the nearest integer is at least $\frac{1}{k+1}$, $\forall~~1\leq j\leq k$. In a recent article Beck, Hosten and Schymura described the Lonely Runner polyhedron and provided a polyhedral approach to identifying families of lonely runner instances. We revisit the Lonely Runner polyhedron and highlight some new families of instances satisfying the conjecture. In addition, we relax the sufficiency of existence of an integer point in the Lonely Runner polyhedron to prove the conjecture. Specifically, we propose that it suffices to show the existence of a lattice point of certain superlattices of the integer lattice in the Lonely Runner polyhedron

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