Eleven, twelve, and thirteen lonely runners

Wills conjectured that, for any non-zero integers $u_1,\dots ,u_k$, there is a real number $t$ such that, for all $i=1,\dots ,k$, $$\parallel t{u}_i\parallel \ge \frac{1}{k+1},$$ where $\parallel x\parallel$ 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 $k\le 9$. We further refine this method with new sieving techniques and employ a polynomial method argument to show that any $(u_1,\dots ,u_k)\equiv (1,2,\dots ,k)(\mathrm{m}\mathrm{o}\mathrm{d}p)$ with $\gcd (u_1,\dots ,u_k)=1$ 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 $k\in \{10,11,12\}$.