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{{harvtxt|Kravitz|2021}} formulated a sharper version of the conjecture that addresses near-equality cases. More specifically, he conjectures that for a given set of speeds <math>v_i</math>, either <math>\delta = s/(s(n-1)+1)</math> for some positive integer <math>s</math>,{{efn|Taking <math>s=1</math> yields the lonely runner conjecture.}} or <math>\delta \geq 1/(n-1)</math>, where <math>\delta</math> is that setup's gap of loneliness. He confirmed this conjecture for <math>n\leq 4</math> and a few special cases.
{{harvtxt|Rifford|2022}} addressed the question of the size of the time required for a runner to get lonely. He formulated a stronger conjecture stating that for every integer <math>n \geq 3</math> there is a positive integer <math>N</math> such that for any collection <math>v_1,v_2,...,v_{n-1}</math> of positive, distinct speeds, there exists some time <math>t>0</math> such that <math>\operatorname{frac}(v_it)\in [1/n,1-1/n]</math> for <math>i=1, ...n-1</math> with
<math display="block">t \leq \frac{N}{\operatorname{min} (v_1,..., v_{n-1})}.</math>
Rifford confirmed this conjecture for <math>n=3,4,5,6</math> and showed that the minimal <math>N</math> in each case is given by <math>N=1</math> for <math>n=3,4,5</math> and <math>N=2</math> for <math>n=6</math>.
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