Lonely runner conjecture

Runners start together on a circular track and keep different constant speeds. With two runners doing one and two laps per unit of time, after half a time unit one is halfway around while the other is back at the start. Their shortest track distance is half a lap. With any finite number of runners, will each eventually have a moment at least one divided by the total runner count of a lap away from every other runner? Distance is measured along the shorter arc of the circle.[2]

Runners with distinct constant speeds start at the same point on a circular track whose circumference is one. Must each eventually be at least $1/n$ of a lap from every other runner, where $n$ is their total number? For the chosen runner $i$, subtract its speed $v_i$ from all speeds. The other runner $j$ completes $t(v_j-v_i)$ relative laps by time $t$. Whole laps bring it back to the chosen runner’s position, so the shortest circular distance is distance to the nearest integer. Write this as $\|x\|=\min_{m\in\mathbb Z}|x-m|$.

The conjecture is that for every finite set of distinct velocities, each $i$ has some $t\ge0$ with $\|t(v_j-v_i)\|\ge1/n$ for all $j\ne i$. The time may depend on $i$; one simultaneous isolation time for all runners is not required. With speeds one and two, taking $t=1/2$ gives a half-integer relative lap and distance one half. The cited preprint states that the real-speed formulation reduces to rational and then integer speed data. Its notation counts $k=n-1$ moving runners relative to a stationary observer, so the threshold is $1/(k+1)$.[2]

The guarantee is known for limited numbers of runners. It has not been proved for every finite number; each runner may use a different moment.[2]

In its positive-integer formulation with at least two moving runners relative to the observer, the September 2026 preprint proves a necessary relation for a counterexample or a case reaching the threshold exactly. There must be integer weights, not all zero, whose weighted sum of the speeds is zero and whose absolute values add to at most $2k+3$. This narrows their structure, rather than excluding every counterexample. The target keeps distinct constant speeds and a common start; letting runners start at different positions changes the problem.[2]

Regular motions can line up again and again, like the hands of several clocks. This question asks whether each motion must also have a time when all the others are sufficiently out of alignment with it. It connects a moving-track picture to the arithmetic of several repeating phases, where a good time must work for all the relative motions together.[2]

For three runners, fix one viewpoint and record the other two fractional lap positions as a point in a square. The point moves as time passes. The chosen runner is isolated by at least one third when both coordinates lie between one third and two thirds, including the boundary. For more runners the square becomes a higher-dimensional box. This geometric view connects the separation question to how an arithmetic motion visits a specified region, rather than merely how often individual runners drift apart.[2]