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arxiv: 2605.27941 · v1 · pith:VMMFSK64new · submitted 2026-05-27 · 🧮 math.NT · math.CO

Mixed thresholds in the Lonely Runner Conjecture

Alathea Jensen This is my paper

Pith reviewed 2026-06-29 10:36 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords lonely runner conjecturemixed thresholdsMLPS_2Fourier seriesarithmetic progressionscircular trackfractional partsDiophantine approximation
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The pith

The set of valid mixed distance thresholds for two runners is exactly characterized.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the lonely runner conjecture by allowing each runner its own distance threshold rather than requiring the same value for all. It defines MLPS_k as the collection of threshold vectors that succeed for every choice of distinct integer speeds. For the case of two runners, an exact description is given of which pairs of thresholds belong to MLPS_2. Fourier series expansions of the relevant indicator functions are developed to produce an arithmetic-progression summation formula and an exact integral expression that support the characterization.

Core claim

For every pair of distinct positive integers s1 and s2 there exists a real t such that the fractional parts satisfy the two inequalities with the given thresholds d1 and d2 precisely when the vector (d1, d2) lies in MLPS_2; the paper supplies the precise membership condition for this set together with the supporting summation identities obtained from the Fourier expansions of the distance-threshold indicators.

What carries the argument

MLPS_2, the set of threshold vectors (d1, d2) for which every pair of distinct positive integer speeds admits a time t at which both runners meet or exceed their respective distances.

Load-bearing premise

The Fourier-series expansions of the distance-threshold indicator functions converge sufficiently to yield an exact arithmetic-progression summation formula without further regularity assumptions on the speeds.

What would settle it

A concrete pair of distinct positive integers s1, s2 together with a vector (d1, d2) that violates the claimed membership condition for MLPS_2, yet for which no t exists satisfying both ||s1 t|| >= d1 and ||s2 t|| >= d2 (or the converse).

read the original abstract

The Lonely Runner Conjecture states that if $k+1$ runners start at the same point on a unit-length circular track and run with distinct constant speeds, then each runner is at some time at least $1/(k+1)$-distant from every other runner. Equivalently, for every tuple of $k$ distinct positive integer speeds $s_1,\ldots,s_k$, there is a real number $t$ such that $\|s_i t\|\geq \frac{1}{k+1}$ for all $i$. We introduce and study a version of the conjecture in which the required distances may vary with $i$. For $\mathbf d=(d_1,\ldots,d_k)\in(0,1/2]^k$, let $\mathsf{MLPS}_k$ be the set of vectors such that, for every choice of distinct positive integer speeds $s_1,\ldots,s_k$, there is a real number $t$ with $\|s_i t\|\geq d_i$ for all $i$. We give an exact characterization of $\mathsf{MLPS}_2$. We also use Fourier series for distance-threshold indicator functions to obtain an arithmetic progression summation formula and an exact two-function integral formula for unequal thresholds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript introduces the mixed lonely runner problem with variable per-runner distance thresholds d = (d1,...,dk) in (0,1/2]^k. It defines MLPS_k as the set of such d for which every tuple of k distinct positive integer speeds admits a time t satisfying ||si t|| >= di for all i. The central claim is an exact characterization of MLPS_2, derived by applying Fourier series to the indicator functions 1_{||x|| >= di} to produce an arithmetic-progression summation formula and an exact two-function integral formula.

Significance. An exact characterization of MLPS_2 would constitute a concrete advance on the lonely runner conjecture by delineating the precise feasible region for unequal thresholds when k=2. The Fourier-analytic approach, if rigorously justified, supplies closed-form tools that could be useful for higher k or for computational verification.

major comments (2)
  1. [Abstract] Abstract: the assertion of an exact arithmetic-progression summation formula obtained 'without additional regularity conditions on the speeds' is load-bearing for the characterization of MLPS_2, yet the derivation from the Fourier series of the discontinuous step-function indicators 1_{||x|| >= di} requires explicit justification that term-by-term summation over the AP remains exact at discontinuity points for arbitrary distinct positive integers s1,...,sk; Carleson's theorem guarantees pointwise a.e. convergence but does not automatically license the interchange with the finite AP sum without further analysis of Gibbs phenomena or sampling of jumps.
  2. [Abstract] Abstract (and the section deriving the integral formula): the claim that the two-function integral formula is exact must be accompanied by a verification that the Fourier coefficients of the product of indicators integrate exactly against the torus measure when the speeds are integers; without this step shown, the characterization of MLPS_2 rests on an unverified interchange.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for identifying points where the Fourier derivations require additional explicit justification. We address each major comment below and will make the necessary revisions to strengthen the rigor of the claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion of an exact arithmetic-progression summation formula obtained 'without additional regularity conditions on the speeds' is load-bearing for the characterization of MLPS_2, yet the derivation from the Fourier series of the discontinuous step-function indicators 1_{||x|| >= di} requires explicit justification that term-by-term summation over the AP remains exact at discontinuity points for arbitrary distinct positive integers s1,...,sk; Carleson's theorem guarantees pointwise a.e. convergence but does not automatically license the interchange with the finite AP sum without further analysis of Gibbs phenomena or sampling of jumps.

    Authors: We agree that the exactness of the arithmetic-progression summation at discontinuity points needs explicit justification beyond the standard a.e. convergence from Carleson's theorem, especially to support the claim for arbitrary distinct positive integer speeds without extra regularity assumptions. The manuscript derives the formula via Fourier series of the indicators but does not include a dedicated verification of the term-by-term interchange for the finite AP sum. We will add a lemma in the revised version that proves the summation remains exact, using the periodicity induced by integer speeds and the finite sum to control the behavior at jumps. revision: yes

  2. Referee: [Abstract] Abstract (and the section deriving the integral formula): the claim that the two-function integral formula is exact must be accompanied by a verification that the Fourier coefficients of the product of indicators integrate exactly against the torus measure when the speeds are integers; without this step shown, the characterization of MLPS_2 rests on an unverified interchange.

    Authors: We acknowledge that the exactness of the two-function integral formula requires an explicit verification that the Fourier coefficients of the product of the indicator functions integrate precisely against the torus measure for integer speeds. The current manuscript states the formula but omits this verification step. We will revise the relevant section to include a direct computation showing that the interchange holds exactly due to the integrality of the speeds, thereby confirming the foundation for the MLPS_2 characterization. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses standard Fourier analysis of periodic functions.

full rationale

The paper's central claim is an exact characterization of MLPS_2 obtained via Fourier-series expansions of the indicator functions 1_{||x|| >= d_i} followed by an arithmetic-progression summation formula and a two-function integral formula. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the derivation invokes only classical Fourier theory on the circle (Carleson pointwise convergence, term-by-term integration over arithmetic progressions) whose validity is independent of the Lonely Runner statement. The work is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard properties of the fractional-part function and Fourier series convergence for indicator functions on the circle; no free parameters or invented entities are introduced.

axioms (2)
  • standard math The fractional-part function ||x|| is dense modulo 1 for irrational multiples of distinct speeds.
    Invoked in the definition of the lonely-runner property for all distinct positive integer speeds.
  • standard math Fourier series of the periodic indicator function of an interval on the circle converge pointwise or in L2 sufficiently to yield exact summation formulas.
    Used to obtain the arithmetic-progression summation and two-function integral formulas.

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Reference graph

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