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Lonely Runner Conjecture (general formulation)

Establish that for every integer k ≥ 1 and every set of k distinct positive integers v1, ..., vk, there exists a real number t such that ||t·vi|| ≥ 1/(k+1) for all indices i ∈ {1, ..., k}.

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Background

The paper proves the conjecture for eight runners (equivalently, for sets of size k=7 in the second formulation), extending the list of values of k for which the conjecture is known. However, the Lonely Runner Conjecture remains open in full generality.

The authors present the conjecture in two equivalent forms and focus on the formulation with positive integer speeds and distance from the origin, which is standard in the literature thanks to equivalence established by Wills. The general resolution of the conjecture for all k is still unresolved.

References

The Lonely Runner Conjecture is a well-known open problem in combinatorial number theory and Diophantine approximation. For all integer k ≥ 1, and every set of distinct positive integers v1, ..., vk, there exists a real number t such that for every i, we have ||tv_i|| ≥ 1/(k+1).

— The lonely runner conjecture holds for eight runners (2509.14111 - Rosenfeld, 17 Sep 2025) in Section 1 (Introduction), Conjecture [Lonely Runner Conjecture] (second formulation)