Lonely runners in function fields

TL;DR

This paper explores a function field analogue of the longstanding lonely runner conjecture, providing partial positive results and extending the problem into a new mathematical setting.

Contribution

It introduces a novel function field analogue of the lonely runner conjecture and proves positive results in certain cases within this new framework.

Findings

Established a function field analogue of the conjecture

Proved the conjecture in specific cases for the new setting

Extended the understanding of the problem into algebraic function fields

Abstract

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.