Large-prime condition for Lemma needOfCovering across all k

Determine whether, for each integer k ≥ 3, there exists a threshold P(k) such that for every prime p ≥ P(k), the verification property in Lemma needOfCovering holds uniformly: namely, for all k-tuples v1, ..., vk ∈ {0, ..., (k+1)p−1} with no vi divisible by p and with gcd(S ∪ {(k+1)p}) = 1 for every (k−1)-element subset S ⊆ {v1, ..., vk}, there exists t ∈ {0, ..., (k+1)p−1} such that ||(t·vi)/((k+1)p)|| ≥ 1/(k+1) for all i.

Background

The core method of the paper uses Lemma needOfCovering to deduce that specific primes must divide the product of speeds in any counterexample to the conjecture, and a computational search verifies this property for many primes p at k=7. The authors conjecture that, ignoring computational cost, for any k a sufficiently large prime p would satisfy the lemma’s conditions, potentially enabling a uniform approach to settle larger k.

Establishing such a large-prime threshold for each k would provide a scalable route to verifying the Lonely Runner Conjecture by reducing the search to finitely many residue classes modulo (k+1)p and would justify the empirical observations reported for k=3–7.