Lonely Runner Conjecture is the following problem, and the conjecture was proven to be true for $k\leq 7$:

Let $V$ be the set of $k$ distinct positive integers with $v_1<v_2<...<v_k$ Then $$\exists t\in [0,1]\quad \forall v_i\in V\quad \|tv_i\|\geq \frac{1}{k+1}$$

where $||x||$ denotes the distance from $x$ to the nearest integer.

It is known that for $k+1$ runners, one can assume the following WLOG:

- $\gcd(v_1,...,v_k)=1$
- $(k+1)|v_1$
- $V$ is neither the set of arithmetic/geometric progression nor prime numbers.

Although it wasn't stated in any paper, it is trivially verifiable that one can also assume the following WLOG:

- $\forall r\in\mathbb{Z}$ s.t. $2\leq r\leq k+1$, $\exists v_i\in V$ s.t. $r|v_i$.
- $\forall r\in\mathbb{Z}$ s.t. $k+2\leq r\leq 2k+1$ and $2|r$, $\exists v_i\in V$ s.t. $r|v_i$.

What can I further assume about $V$?

The first three assumptions were stated in View-obstruction: a shorter proof for 6 lonely runners by Jérôme Renault, which is also a nice introduction to LRC along with the corresponding Wikipedia page (Lonely runner conjecture). The lonely runner with seven runners by Barajas is also a relevant reference.

Not-lonely runners. Will Sawin's answer establishes only "logarithmic distance between upper and lower bounds." $\endgroup$