Related papers: Mixed thresholds in the Lonely Runner Conjecture

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…

The Lonely Runner Conjecture of Wills and Cusick states that if $k+1$ runners start running at distinct constant speeds around a unit-length circular track, then for each runner there is a time when he/she is at least $1/(k+1)$ away from…

We introduce a sharpened version of the well-known Lonely Runner Conjecture of Wills and Cusick. Given a real number $x$, let $\Vert x \Vert$ denote the distance from $x$ to the nearest integer. For each set of positive integer speeds $v_1,…

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, 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…

The Lonely Runner Conjecture asserts that if $n$ runners with distinct constant speeds run on the unit circle $\mathbb{R}/\mathbb{Z}$ starting from $0$ at time $0$, then each runner will at some time $t>0$ be lonely in the sense that she/he…

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 ${\bf R}/{\bf Z}$ starting at a common time and place, then each runner…

Wills conjectured that, for any non-zero integers $u_1,\ldots,u_k$, there is a real number $t$ such that, for all $i=1,\ldots,k$, \[\lVert tu_i\rVert\geq\frac{1}{k+1},\] where $\lVert x\rVert$ is the distance from $x$ to the closest…

We study the \emph{Lonely Runner Conjecture}, conceived by J\"org M.~Wills in the 1960's: Given positive integers $n_1, n_2, \dots, n_k$, there exists a positive real number $t$ such that for all $1 \le j \le k$ the distance of $t \, n_j$…

The lonely runner conjecture of Wills and Cusick asserts that if $n$ runners with distinct constant speeds run around a a circular unit length track, starting at a common time and place, then each runner will at some time be separated by a…

In this paper, we study the distribution of the boundary points of expansion. As an application, we say something about the lonely runner problem. We show that given $k$ runners $\mathcal{S}_i$ round a unit circular track with the condition…

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…

The Lonely Runner Conjecture is a number theory problem, dating to 1964. Using dynamical systems theory, we show almost all sets of velocities solve the conjecture. Furthermore, any "traditional" approach of Diophantine approximation cannot…

Let $||x||$ be the absolute distance from $x$ to the nearest integer. For a set of distinct positive integral speeds $v_1, \ldots, v_n$, we define its maximum loneliness, also known as the gap $\delta$, to be $$ML(v_1,\ldots,v_n) = \max_{t…

Tao (2018) showed that in order to prove the Lonely Runner Conjecture (LRC) up to $n+1$ runners it suffices to consider positive integer velocities in the order of $n^{O(n^2)}$. Using the zonotopal reinterpretation of the conjecture due to…

We prove that the lonely runner conjecture holds for eight runners. Our proof relies on a computer verification and on recent results that allow bounding the size of a minimal counterexample. We note that our approach also applies to the…

We prove that the lonely runner conjecture holds for nine runners. Our proof is based on a couple of improvements of the method we used to prove the conjecture for eight runners.

We consider (n+1) runners with given constant unique integer speeds running along the circumference of a circle whose circumferential length is one, and all runners starting from the same point. We define and give lower bounds to a first…

For each subtorus $T$ of $(\mathbb{R}/\mathbb{Z})^n$, let $D(T)$ denote the (infimal) $L^\infty$-distance from $T$ to the point $(1/2,\ldots, 1/2)$. The $n$-th Lonely Runner spectrum $\mathcal{S}(n)$ is defined to be the set of all values…

The Lonely Runner Conjecture originated in Diophantine approximation is turning 60. Even if the conjecture is still widely open, the flow of partial results, innovative tools and connections to different problems and applications has been…