Eleven, twelve, and thirteen lonely runners

TL;DR

This paper advances the proof of the Lonely Runner Conjecture for up to 12 runners by refining computational methods and employing new sieving and polynomial techniques.

Contribution

It introduces refined sieving and polynomial methods to verify the conjecture for specific cases up to 12 runners, extending previous computational verifications.

Findings

Verified the conjecture for k=10, 11, 12 using computer-assisted proofs.

Developed new sieving techniques to improve computational verification.

Applied polynomial methods to establish the conjecture under certain modular conditions.

Abstract

Wills conjectured that, for any non-zero integers $u_1,\ldots,u_k$, there is a real number $t$ such that, for all $i=1,\ldots,k$, \[\lVert tu_i\rVert\geq\frac{1}{k+1},\] where $\lVert x\rVert$ is the distance from $x$ to the closest integer. This statement is known as the Lonely Runner Conjecture. A computational method developed by Rosenfeld and the second author verified the conjecture for $k\leq9$. We further refine this method with new sieving techniques and employ a polynomial method argument to show that any $(u_1,\ldots,u_k)\equiv(1,2,\ldots,k)\pmod{p}$ with $\gcd(u_1,\ldots,u_k)=1$ satisfies the conjecture when $k+1$ and $p > k^2+k$ are both odd primes. Ultimately, we provide a computer-assisted proof of the Lonely Runner Conjecture for $k\in\{10,11,12\}$.