Explicit-time Lonely Runner Conjecture

Develop, for each integer n ≥ 1 and any n-set V of positive integer speeds, a universal positive integer M such that at the prescribed rational time t = M/(2^{⌈ln 2 v_{n+1}⌉}(n+1)v_n), the minimum distance of tv to the nearest integer across v ∈ V is at least 1/(n+1).

Background

This conjecture proposes explicit times, based on logarithmic functions of the largest speeds, at which loneliness occurs, supported by computational simulations. It aims to make the existence claim constructive and uniform across speed sets.

If established, it would provide a practical method to witness loneliness at specific rational times with bounded denominators.