Lonely Runners over Function Fields: Quantized Phase--Riesz product

Abstract: Let $C_k(q)$ be the least cardinality of a family of nonzero polynomials over $\mathbb F_q$ whose associated codimension-$k$ partial-circulant kernels cover the full coefficient space. Chow and Rimani'c conjectured that $C_k(q)=1+q+\cdots+q<sup>k$. We disprove the unrestricted conjecture by constructing thirteen monic polynomials over $\mathbb F_2$ whose $k=3$ kernels cover $\mathbb F_2<sup>7$; in particular, $C_3(2)\le 13&lt;15$. For a general covering family of size $N=q<sup>k+S$ and $\mathbb F_q$-linear rank $d$, we prove $$S\gg d<sup>{2/3}\left(\frac{\log(2q)}{\log(eNq<sup>k/S)}\right)<sup>{2/3}$$. Consequently, for every fixed $k\ge 2$ and all sufficiently large $q$, $C_k(q)\ge q<sup>k+c_kq<sup>{2/3}$. When $k=2$, an integer-multiplicity refinement of the second-moment covering argument yields $$\liminf_{q\to\infty}(C_2(q)-q^2)/q\ge \widetilde c_2$$, where $\widetilde c_2$ is an explicit one-variable variational constant with numerical value $\widetilde c_2=0.5829944375\ldots$. We also classify triples admitting two independent low-degree polynomial syzygies and prove a conditional packet-free lower bound of size $q<sup>k+(1/2-o(1))q<sup>{k-1}$.

• The paper disproves the unrestricted Chow–Rimanić conjecture by constructing 13 polynomials over \(\mathbb{F}_2\) with codimension-three kernels covering \(\mathbb{F}_2^7\), giving \(C_3(2)\le13<15\).
• A quantized \(q\)-ary phase–Riesz product proves that any covering family of rank \(d\) and size \(q^k+S\) has excess \(S\gg d^{2/3}\), leading for fixed \(k\) to \(C_k(q)\ge q^k+c_kq^{2/3}\) for sufficiently large \(q\).
• An integer-multiplicity inequality raises the rigorous \(k=2\) asymptotic constant above \(0.5278\) from \(0.4877\), while syzygy classification identifies packet structures obstructing the conjectural \(q^{k-1}\) improvement.

The function-field lonely runner problem

The Lonely Runner Conjecture asks whether $n$ runners with distinct constant speeds on a unit circle each attain circular distance at least $1/(n+1)$ from every other runner at some common time. After reducing to integer speeds and fixing one runner at the origin, the quantity of interest is $\operatorname{ML}(V)=\sup_t \min_v \|tv\|$, conjectured to be at least $1/(n+1)$. Chow and Rimanić introduced a non-Archimedean analogue: for a finite family $\mathcal{F}\subset \mathbb{F}_q[T]\setminus\{0\}$, the loneliness is $$\delta(\mathcal{F})=\sup_{\alpha}\min_f \|\alpha f\|$$, where $\|\cdot\|$ is the fractional-part norm on $\mathbb{F}_q((T^{-1}))$. Because the norm is discrete, the natural threshold is $q^{-k}$, and the covering number

$$C_k(q):=\min\{|\mathcal{F}|:\delta(\mathcal{F})<q^{-k}\}$$

satisfies the elementary bounds $1/(n+1)$0. The Chow–Rimanić conjecture asserts equality with $1/(n+1)$1; their own work proved this under a small-degree hypothesis and gave $1/(n+1)$2 via sunflower geometry.

The paper under review makes three contributions: it disproves the unrestricted conjecture by an explicit counterexample over $1/(n+1)$3; it proves a rank-sensitive lower bound $1/(n+1)$4 on the covering excess $1/(n+1)$5 using a quantized $1/(n+1)$6-ary phase–Riesz product; and it sharpens the $1/(n+1)$7 estimate through an integer-multiplicity inequality, yielding a rigorous improvement of the Chow–Rimanić constant. A classification of low-degree polynomial syzygies identifies the first obstruction to pushing the fixed-$1/(n+1)$8 bound to the natural scale $1/(n+1)$9.

The covering formulation and exact Fourier support

The reduction to linear algebra proceeds as follows. For speeds of degree at most $\operatorname{ML}(V)=\sup_t \min_v \|tv\|$0, let $\operatorname{ML}(V)=\sup_t \min_v \|tv\|$1 with the coefficient pairing, and attach to each speed $\operatorname{ML}(V)=\sup_t \min_v \|tv\|$2 the subspace $\operatorname{ML}(V)=\sup_t \min_v \|tv\|$3 (where $\operatorname{ML}(V)=\sup_t \min_v \|tv\|$4) and its annihilator $\operatorname{ML}(V)=\sup_t \min_v \|tv\|$5, a codimension-$\operatorname{ML}(V)=\sup_t \min_v \|tv\|$6 "partial-circulant kernel." Then $\operatorname{ML}(V)=\sup_t \min_v \|tv\|$7 holds exactly when the kernels cover $\operatorname{ML}(V)=\sup_t \min_v \|tv\|$8. The decisive structural feature is the exact Fourier identity

$\operatorname{ML}(V)=\sup_t \min_v \|tv\|$9

where $1/(n+1)$0 is the covering multiplicity. Every Fourier coefficient is nonnegative, every nonzero coefficient is at least $1/(n+1)$1, and the Fourier support is the finite union $1/(n+1)$2. Positivity, quantization, and finite support are used simultaneously in the main argument—a simplification unavailable in the real-circle setting, where Bohr-set Fourier coefficients carry signs and lack a fixed quantum.

A counterexample over $1/(n+1)$3

The unrestricted conjecture fails in characteristic two. The author exhibits thirteen monic polynomials of degree at most four whose codimension-three kernels cover $1/(n+1)$4, so that $1/(n+1)$5, with loneliness exactly $1/(n+1)$6. The certificate is an exhaustive check of all $1/(n+1)$7 coefficient vectors, reproducible by a short standard-library Python script; the multiplicity distribution has no zero entries, and the total incidence count $1/(n+1)$8 matches the sum of kernel sizes. The example was located computationally via a deep cross-entropy search in the spirit of Wagner's neural-network constructions, then verified directly. This result does not conflict with the large-$1/(n+1)$9 theorems below: the failure is confined to small pairs $\mathcal{F}\subset \mathbb{F}_q[T]\setminus\{0\}$0, and the author proposes as a corrected statement that $\mathcal{F}\subset \mathbb{F}_q[T]\setminus\{0\}$1 once $\mathcal{F}\subset \mathbb{F}_q[T]\setminus\{0\}$2.

Quantized $\mathcal{F}\subset \mathbb{F}_q[T]\setminus\{0\}$3-ary phase–Riesz products

The central estimate is sensitive to the $\mathcal{F}\subset \mathbb{F}_q[T]\setminus\{0\}$4-linear rank $\mathcal{F}\subset \mathbb{F}_q[T]\setminus\{0\}$5: any covering family of size $\mathcal{F}\subset \mathbb{F}_q[T]\setminus\{0\}$6 satisfies

$\mathcal{F}\subset \mathbb{F}_q[T]\setminus\{0\}$7

with absolute implied constant. The proof adapts Bedert's Riesz-product strategy to the function-field setting. Choosing independent speeds $\mathcal{F}\subset \mathbb{F}_q[T]\setminus\{0\}$8, the surjective map $\mathcal{F}\subset \mathbb{F}_q[T]\setminus\{0\}$9 produces fibre densities $$\delta(\mathcal{F})=\sup_{\alpha}\min_f \|\alpha f\|$$0, and the averaged excess $$\delta(\mathcal{F})=\sup_{\alpha}\min_f \|\alpha f\|$$1 is nonnegative with mean $$\delta(\mathcal{F})=\sup_{\alpha}\min_f \|\alpha f\|$$2 and Fourier coefficients exactly $$\delta(\mathcal{F})=\sup_{\alpha}\min_f \|\alpha f\|$$3. Wolff's sharp hypercontractivity theorem for finite probability spaces—reducing to the two-point space with smallest atom $$\delta(\mathcal{F})=\sup_{\alpha}\min_f \|\alpha f\|$$4—yields the level bound

$$\delta(\mathcal{F})=\sup_{\alpha}\min_f \|\alpha f\|$$5

where $$\delta(\mathcal{F})=\sup_{\alpha}\min_f \|\alpha f\|$$6. Testing $$\delta(\mathcal{F})=\sup_{\alpha}\min_f \|\alpha f\|$$7 against the negative product built from $$\delta(\mathcal{F})=\sup_{\alpha}\min_f \|\alpha f\|$$8 gives the master inequality $$\delta(\mathcal{F})=\sup_{\alpha}\min_f \|\alpha f\|$$9; optimizing in $\|\cdot\|$0 yields the theorem. Two features deserve emphasis. First, the factor $\|\cdot\|$1 in the $\|\cdot\|$2-ary level estimate is cancelled by the coefficient $\|\cdot\|$3 in the negative product, which removes the logarithmic loss that a binary or Steinhaus phase would incur. Second, the exponent $\|\cdot\|$4 is intrinsic to balancing the three terms ($\|\cdot\|$5); improving it requires input beyond rank and Fourier quantization.

To convert the rank-sensitive bound into a uniform one, low-rank covers must be excluded. An arbitrary vector-space compression would destroy the blocks $\|\cdot\|$6, so the author instead reduces modulo a monic irreducible polynomial of degree $\|\cdot\|$7, working inside $\|\cdot\|$8 where multiplication by $\|\cdot\|$9 is respected. Iterating this step compresses any rank-$\mathbb{F}_q((T^{-1}))$0 cover into degree range below $\mathbb{F}_q((T^{-1}))$1 while preserving size and rank. The small-degree theorem of Chow–Rimanić then forces $\mathbb{F}_q((T^{-1}))$2 for large $\mathbb{F}_q((T^{-1}))$3, and the phase–Riesz estimate yields:

Fixed-$\mathbb{F}_q((T^{-1}))$4 lower bound: for every fixed $\mathbb{F}_q((T^{-1}))$5 there are $\mathbb{F}_q((T^{-1}))$6 and $\mathbb{F}_q((T^{-1}))$7 such that $\mathbb{F}_q((T^{-1}))$8 for all prime powers $\mathbb{F}_q((T^{-1}))$9.

This remains short of the natural second term $q^{-k}$0; the gap is addressed structurally in the syzygy analysis below.

Integer-multiplicity refinement at $q^{-k}$1

For $q^{-k}$2 a separate argument improves the constant on the linear scale. The new input is a pointwise inequality valid only because covering multiplicities are integers: for $q^{-k}$3,

$q^{-k}$4

which follows from the integrality of $q^{-k}$5. Unlike Cauchy–Schwarz, this uses the full second factorial moment optimally; maximizing over $q^{-k}$6 produces the piecewise-linear envelope $q^{-k}$7, the interpolation of $q^{-k}$8 at the integers, which dominates $q^{-k}$9 strictly off the lattice. Inserting $$C_k(q):=\min\{|\mathcal{F}|:\delta(\mathcal{F})<q^{-k}\}$$0 into both the generic-intersection and sunflower branches of the Chow–Rimanić ordering yields improved envelopes $$C_k(q):=\min\{|\mathcal{F}|:\delta(\mathcal{F})<q^{-k}\}$$1 and $$C_k(q):=\min\{|\mathcal{F}|:\delta(\mathcal{F})<q^{-k}\}$$2, and hence a variational lower bound

$$C_k(q):=\min\{|\mathcal{F}|:\delta(\mathcal{F})<q^{-k}\}$$3

with numerical value $$C_k(q):=\min\{|\mathcal{F}|:\delta(\mathcal{F})<q^{-k}\}$$4 attained near $$C_k(q):=\min\{|\mathcal{F}|:\delta(\mathcal{F})<q^{-k}\}$$5. A rigorous elementary argument gives $$C_k(q):=\min\{|\mathcal{F}|:\delta(\mathcal{F})<q^{-k}\}$$6, improving the previous constant $$C_k(q):=\min\{|\mathcal{F}|:\delta(\mathcal{F})<q^{-k}\}$$7 unconditionally; the decimal approximation plays no role in the exact variational statement. Since $$C_k(q):=\min\{|\mathcal{F}|:\delta(\mathcal{F})<q^{-k}\}$$8 already extracts the optimal consequence of the first two factorial moments, any further improvement must exploit higher moments or additional geometry—the paper sketches a stability dichotomy between an energy-dispersed regime (where a third factorial moment helps) and a structured regime (where intersections form countable packets).

Syzygies and the conditional packet-free bound

Higher moments of $$C_k(q):=\min\{|\mathcal{F}|:\delta(\mathcal{F})<q^{-k}\}$$9 detect tuples with unexpectedly small block rank: the exact identity $1/(n+1)$00 shows that near-union-bound covers force many such tuples. Rank defects translate into polynomial syzygies $1/(n+1)$01 with coefficients in $1/(n+1)$02. The paper classifies triples admitting two independent syzygies: either there is a nonzero relation with all three coefficients of degree at most $1/(n+1)$03, or the triple shares a common factor $1/(n+1)$04 with quotients of degree at most $1/(n+1)$05. These are precisely the "packets" that inflate triple intersections by a factor $1/(n+1)$06 and obstruct second-moment arguments at the $1/(n+1)$07 scale.

Under an explicit quantitative hypothesis—that after a suitable kernel ordering, all but $1/(n+1)$08 kernels admit about $1/(n+1)$09 distinct generic intersections among which harmful pairs have negligible second-moment mass—the author proves the conditional bound $1/(n+1)$10. The hypothesis is genuinely restrictive: locally labelled relations need not assemble into globally coherent packets, since labels vary across overlapping triples, scalar multiples describe the same projective relation, and moment numerology can be reproduced without any common factor or recurrence. Removing this assumption amounts to a labelled local-to-global agreement theorem, which the paper identifies as the function-field counterpart of the relation-compatibility obstruction in Tao's higher-moment approach to the classical conjecture. The advantage here is that labels lie in the fixed space $1/(n+1)$11 and the first obstruction admits the explicit two-type classification above.

Limitations and open questions

Several caveats bear directly on the results. The counterexample settles only $1/(n+1)$12; the exact value and the classification of all small-pair failures remain open. The corrected large-field conjecture $1/(n+1)$13 for $1/(n+1)$14 is stated but unproved, and even its weaker target $1/(n+1)$15 is out of reach: at $1/(n+1)$16 the proven proportion is $1/(n+1)$17, and for $1/(n+1)$18 no positive multiple of $1/(n+1)$19 is known. The exponent $1/(n+1)$20 is hard-wired into the master inequality's quadratic structure, and the proposed remedies—showing Fourier multiplicities usually exceed one, or exploiting dependent directions—are not carried out. Block-level phases attaching coordinates to the full spaces $1/(n+1)$21 would record relations directly but destroy automatic normalization; quotienting or correlated phase laws are suggested but undeveloped. Finally, the growing-$1/(n+1)$22 regime lacks a uniform theory, since both the compression endpoint and the packet analysis depend on fixed $1/(n+1)$23.

Conclusion

This paper reframes the function-field lonely runner problem around three exact structures—partial-circulant covering, quantized nonnegative Fourier support, and integer-valued multiplicity—and exploits each decisively. It refutes the unrestricted Chow–Rimanić conjecture with a verifiable thirteen-kernel cover over $1/(n+1)$24, establishes the first power-saving lower bounds $1/(n+1)$25 valid for all sufficiently large fields, and rigorously improves the $1/(n+1)$26 constant from $1/(n+1)$27 to beyond $1/(n+1)$28 via an integer-multiplicity inequality. The explicit syzygy classification isolates the algebraic obstruction separating these results from the conjectural $1/(n+1)$29 scale, reducing progress to a well-defined labelled agreement problem.