  Let $\|x\|$ be the absolute distance from $x$ to the nearest integer. For a
set of distinct positive integral speeds $v_1, \ldots, v_n$, we define its
maximum loneliness to be
  $$\text{ML}(v_1,\ldots,v_n) = \max_{t \in \mathbb{R}}\min_{1 \leq i \leq n}
\|tv_i\|$$
  The Loneliness Spectrum Conjecture, recently proposed by Kravitz, asserts
that
  $$\exists s \in \mathbb{N}, \text{ML}(v_1,\ldots,v_n) = \frac{s} {sn + 1}
\text{ or } \text{ML}(v_1,\ldots,v_n) \ge \frac{1}{n}$$
  We disprove the Loneliness Spectrum Conjecture for $n = 4$ and propose an
alternative conjecture. We confirm the amended conjecture for $n = 4$ if any
pair of speeds share a common factor of at least $3$ and also prove some
related results.
