Suppose k runners having distinct constant speeds start at a common point and run laps on a unit length circular track

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Identify the unit-length circular track with $\mathbb R/\mathbb Z$. For a real number $u$, write $\|u\|_{\mathbb R/\mathbb Z}=\min_{m\in\mathbb Z}|u-m|$ for its circular distance from zero. Let $k\ge2$ and let $v_1,\ldots,v_k$ be distinct real constant velocities. With all runners starting at the same point at time zero, runner $i$ has position $tv_i$ modulo one at time $t\ge0$.

Statement recorded in the supplied source.

For every such choice of velocities and each $i\in\{1,\ldots,k\}$, is there a time $t\ge0$ such that \[\|t(v_j-v_i)\|_{\mathbb R/\mathbb Z}\ge\frac1k\qquad\text{for every }j\ne i?\] The time may depend on the runner. For one runner the assertion is vacuous. [S2]

Short English statement

Does every runner eventually have circular distance at least the reciprocal of the number of runners from every other runner?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3100057 (AMR-030-0057). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 J. Barajas and O. Serra, The lonely runner with seven runners (2007), Abstract and Section1, geometric Lonely Runner Conjecture and nearest-integer reformulation, pages1--2. https://arxiv.org/abs/0710.4495.