Correlation among runners and some results on the Lonely Runner Conjecture

TL;DR

This paper investigates the Lonely Runner Conjecture by analyzing correlations among runners' proximity to the origin, improving existing bounds, extending previous results, and introducing dynamic interval graphs to explore a weaker version of the conjecture.

Contribution

It introduces correlation analysis among runners, improves bounds on loneliness gaps, extends previous results, and proposes dynamic interval graphs for a weaker conjecture version.

Findings

Improved bounds on the gap of loneliness.

Extended results on invisible runners.

Introduced dynamic interval graphs for the conjecture.

Abstract

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 along the unit circle with constant different speeds and starting at the same point, there is a moment where all of them are far enough from the origin. We study the correlation among the time that runners spend close to the origin. By means of these correlations, we improve a result of Chen on the gap of loneliness and we extend an invisible runner result of Czerwinski and Grytczuk. In the last part, we introduce dynamic interval graphs to deal with a weak version of the conjecture thus providing some new results.