THE LONELY RUNNER FINALLY GETS SOME ELBOW ROOM
Imagine a circular track one lap long. Several runners start from the same point at the same moment, each at its own constant speed, all different. The lonely runner conjecture says that every runner will, at some point, find itself alone: at a distance of at least 1/(n + 1) of a lap from all the others, when there are n + 1 runners in total.
The problem is older than its name. According to Xiyu Hu (University of Chinese Academy of Sciences) and Yifan Jing (Ohio State University), it goes back to the 1965 doctoral thesis of Jörg Wills on the approximation of numbers by fractions. Cusick gave an equivalent version in terms of lines of sight blocked by obstacles; the runners came later, with a reformulation by Goddyn. It has since been linked to billiard trajectories, colourings of graphs and flows in abstract structures called matroids.
Measuring loneliness
Mathematicians usually fix one runner and look at the others’ speeds relative to it: a set V of n distinct positive whole numbers. At time t, a runner with speed v sits at position vt around the track, and its distance from the start is written ∥vt∥ — the distance from vt to the nearest whole number. The maximum loneliness of V is
ML(V) = max over t of min over v in V of ∥vt∥
The best moment, judged by the closest runner. The conjecture claims ML(V) ≥ 1/(n + 1), and that would be sharp: for the speeds 1, 2, …, n, the maximum loneliness is exactly 1/(n + 1). With two moving runners at speeds 1 and 2, for instance, the fixed runner can be at most a third of a lap from both — at time 1/3.
The easy half
Half of the answer comes cheaply. The times when a given runner is within δ of the start fill a fraction 2δ of the clock. With n runners, if 2nδ is less than 1, these time windows cannot cover everything, so some moment is free of all of them. Letting δ grow to its limit gives
ML(V) ≥ 1/(2n)
That is roughly half of what the conjecture predicts. Getting beyond it has proved stubborn. The authors list the history: for decades, gains of order 1/n² only; then improvements by Perarnau and Serra, by Terence Tao in 2018, by Benjamin Bedert in 2025. But all of them share a weakness: relative to 1/(2n), the improvement tends to zero as the number of runners grows.
A gain that does not fade
The new theorem changes that:
ML(V) > (1 + c)/(2n)
for a fixed constant c > 0 and every set of n ≥ 2 speeds. The extra margin is now a constant fraction of the easy bound, however many runners there are. The constant is explicit but not computed in the paper, and the authors make no attempt to optimise it. It does not reach the conjectured 1/(n + 1); it is the first gain that does not dissolve.
The proof looks at how the “danger windows” of different runners overlap. Some overlap because they share a centre — runners whose speeds have common divisors hit the start together. Others overlap at different centres. For the first kind, the authors refine an argument about large common divisors due to Green and Walker. For the second, they prove a new geometric inequality: when the windows are widened around their centres, their overlaps must grow faster than in proportion. Either way, the uncovered time cannot shrink to nothing.
As a bonus, the result improves a lower bound for the “pyjama problem”: how many rotated and stretched copies of a striped pattern are needed to cover the plane.
Who did what
The paper carries an explicit statement on AI use. The strategy and framework were the authors’ own. ChatGPT 6 Pro and ChatGPT 6 Astra were used mainly to check arguments, turn informal proofs into rigorous ones and explore refinements. Pro suggested bringing in sums over common divisors, and later fixed “some small gaps” in a first version of the key inequality’s proof. The final proofs were written by the authors. Their verdict on the tools is precise: “the models were unable to improve the lonely runner bound on their own.”
Sixty years after Wills, the loneliness each runner is guaranteed has, for the first time, a margin that stays put.