Function field Lonely Runner Conjecture

Establish that for every prime power q and every nonzero polynomial set F ⊂ Fq[X] with size satisfying 1 ≤ |F| < (q^{k+1} − 1)/(q − 1) for some integer k ≥ 0, the function-field loneliness gap satisfies κq(F) ≥ q^{−k}, where κq(F) is the supremum over α in the unit ‘torus’ T of the minimal absolute value |f α| across f ∈ F.

Background

This q-analog reformulates the Lonely Runner phenomenon in the setting of Laurent series over finite fields, replacing distances to integers by valuations and measuring gaps via q-adic norms. The upper bound on |F| is necessary, and the conjecture has been proved for some small sizes and bounded degree cases.

The conjecture connects discrete structures in function fields with Diophantine approximation analogs, aiming for parallels to the real/integer versions.

References

Conjecture 31 (Function field LR Conjecture). For every prime power q and every F ⊂ F [q] \ {0} set of polynomials with k+1 q − 1 ≤ |F| < q − 1 for some k ∈ N,

we have that −k κq(F) ≥ q .

— The Lonely Runner Conjecture turns 60  (2409.20160 - Perarnau et al., 2024) in Conjecture 31, Section 10.6 (Function fields)

The counterexample in \cref{sec:counterexample} rules out the original unrestricted formulation, but it remains natural to ask whether the projective threshold is correct once the coefficient field is sufficiently large.

— Lonely Runners over Function Fields: Quantized Phase--Riesz product  (2608.18818 - Hu, 19 Aug 2026) in Section 6, “Further questions,” subsection “A corrected large-field conjecture”