The lonely runner with seven runners

Suppose $k+1$ runners having nonzero constant speeds run laps on a unit-length circular track starting at the same time and place. A runner is said to be lonely if she is at distance at least $1/(k+1)$ along the track to every other runner. The lonely runner conjecture states that every runner gets lonely. The conjecture has been proved up to six runners ($k\le 5$). A formulation of the problem is related to the regular chromatic number of distance graphs. We use a new tool developed in this context to solve the first open case of the conjecture with seven runners

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

The lonely runner problem for many runners

The lonely runner conjecture asserts that for any positive integer n and any positive numbers v1 < ... < vn there exists a positive number t such that ||vi t|| ≥ 1/(n+1) for every i=1, ...,n. We verify this conjecture for n ≥ 16342 under assumption that the speeds of the runners satisfy vj+1/vj ≥ 1+33 log n/n for j=1, ...,n-1

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