Characterize all tight instances of the Lonely Runner Conjecture

Classify all sets of n positive integer speeds V for which the loneliness gap attains equality, i.e., κ(V) = 1/(n+1), modulo dilation. Identify necessary and sufficient arithmetic conditions describing precisely which speed sets are tight.

Background

Tight instances are sets of speeds for which the conjectured bound is met exactly. Beyond dilations of [n], several sporadic and infinite families are known via specific arithmetic constructions (e.g., accelerating selected runners).

Despite progress, a general characterization remains elusive and is linked to deep number-theoretic problems such as Jacobsthal’s problem.

References

The problem of providing a complete characterization of tight instances is still widely open. In particular, the converse of Theorem 12 does not hold in its full generality; see [44, Section 3].

— The Lonely Runner Conjecture turns 60  (2409.20160 - Perarnau et al., 2024) in Section 4 (Tight Instances)

Does $\max_iv_i<3q$ hold for every near-tight sextuple? More generally, Kravitz has pointed out that a bound $\max_iv_i\le C(n)\,q$ for all but finitely many near-tight $n$-tuples follows from whenever the Lonely Runner Conjecture holds for $n-1$ and $n-2$ speeds, the constant being a maximum over the finitely many critical subtori; the homogeneity argument in the proof of Theorem \ref{thm:UC} is how such a constant is computed. Its value for other $n$, and whether the sporadic tuples respect it, are open.

— Odd denominators in the Lonely Runner spectrum for six speeds  (2609.03444 - Cordella, 3 Sep 2026) in Question 2, Section 4.4, “Questions”