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arxiv: 2306.10417 · v2 · submitted 2023-06-17 · 🧮 math.NT · math.CO

Amending the Lonely Runner Spectrum Conjecture

Ho Tin Fan , Alec Sun This is my paper

Pith reviewed 2026-05-24 08:22 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords lonely runnermaximum lonelinessspectrum conjecturecounterexamplesinteger speedsdiophantine approximationnumber theory
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The pith

The loneliness spectrum conjecture fails for four speeds via an infinite family of counterexamples.

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

The paper shows that the Loneliness Spectrum Conjecture does not hold when there are four distinct positive integer speeds. It constructs infinitely many such sets whose maximum loneliness lies strictly between the two possibilities the conjecture allows. An amended conjecture is proposed to cover the intermediate values that appear, and the amended form is proved whenever any two speeds share a common factor of three or larger.

Core claim

The Loneliness Spectrum Conjecture asserts that for any n distinct positive integral speeds the maximum loneliness ML is either equal to s/(s n + 1) for some natural number s or else at least 1/n. For n = 4 this statement is false: there exist infinitely many four-speed sets for which ML is a number strictly larger than every s/(4s + 1) yet strictly smaller than 1/4. The authors replace the original conjecture with an amended version that accounts for these intermediate loneliness values and prove the amended version under the arithmetic condition that some pair of speeds has gcd at least 3.

What carries the argument

The maximum-loneliness function ML(v1,…,vn) = max_t min_i ||t v_i||, which records the largest simultaneous distance any real time t can achieve from all the runners to the nearest integer.

If this is right

  • The original Loneliness Spectrum Conjecture is false for every set of four speeds belonging to the constructed infinite family.
  • An amended conjecture is stated that permits loneliness values strictly between s/(4s+1) and 1/4.
  • The amended conjecture holds for all four-speed sets in which at least one pair of speeds has a common factor of three or more.
  • Additional related results on the possible values of ML are established for four speeds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the amended conjecture is true in full generality, the possible values of ML for four speeds become completely determined by the arithmetic properties of the speed set.
  • Analogous intermediate loneliness values may appear for n greater than 4, indicating that the original spectrum statement requires amendment beyond the n=4 case.

Load-bearing premise

The explicit constructions of the infinite family of four-speed sets produce a maximum loneliness value strictly between s/(4s+1) and 1/4, and this value is correctly computed from the definition of ML without hidden dependencies on the conjecture itself.

What would settle it

Direct computation of ML for one concrete quadruple from the infinite family (for example by determining the supremum of the piecewise-linear min-distance function over one period) that yields a numerical value lying strictly between some s/(4s+1) and 1/4.

read the original abstract

Let $||x||$ be the absolute distance from $x$ to the nearest integer. For a set of distinct positive integral speeds $v_1, \ldots, v_n$, we define its maximum loneliness, also known as the gap $\delta$, to be $$ML(v_1,\ldots,v_n) = \max_{t \in \mathbb{R}}\min_{1 \leq i \leq n} || tv_i||.$$ The Loneliness Spectrum Conjecture, recently proposed by Kravitz (2021), asserts that $$\exists s \in \mathbb{N}, \text{ML}(v_1,\ldots,v_n) = \frac{s} {sn + 1} \text{ or } \text{ML}(v_1,\ldots,v_n) \geq \frac{1}{n}. $$ We disprove the Loneliness Spectrum Conjecture for $n = 4$ with an infinite family of counterexamples and propose an alternative conjecture. We confirm the amended conjecture for $n = 4$ whenever there exists a pair of speeds with a common factor of at least $3$ and also prove some related results.

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

1 major / 1 minor

Summary. The paper disproves the Loneliness Spectrum Conjecture for n=4 via an infinite family of explicit 4-tuples of speeds where ML lies strictly between s/(4s+1) and 1/4, proposes an alternative conjecture, and proves the amended version for n=4 under the condition that some pair of speeds shares a common factor of at least 3.

Significance. If the constructions and ML computations hold, the work supplies the first counterexamples to the conjecture together with a conditional proof of a refined statement, advancing the lonely-runner spectrum problem in Diophantine approximation.

major comments (1)
  1. [the section presenting the infinite family of counterexamples] The section presenting the infinite family of counterexamples: the claim that ML(v) lies strictly between s/(4s+1) and 1/4 is load-bearing for the disproof and requires explicit verification that the supremum of min_i ||t v_i|| equals an interior value; the manuscript must show both the lower bound (no t achieves a larger min) and that the value never equals any s/(4s+1).
minor comments (1)
  1. The abstract states the amended conjecture only implicitly; a one-sentence formulation of the proposed alternative would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for more explicit verification in the counterexample section. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [the section presenting the infinite family of counterexamples] The section presenting the infinite family of counterexamples: the claim that ML(v) lies strictly between s/(4s+1) and 1/4 is load-bearing for the disproof and requires explicit verification that the supremum of min_i ||t v_i|| equals an interior value; the manuscript must show both the lower bound (no t achieves a larger min) and that the value never equals any s/(4s+1).

    Authors: We agree that the disproof requires fully explicit verification of both the achieved supremum and its strict position between the indicated thresholds. The current manuscript constructs the infinite family via explicit 4-tuples (with speeds scaled by a parameter k) and states the resulting ML value, but the lower-bound argument (via an explicit t attaining the min) and the strict inequality (via denominator comparison or interval separation from s/(4s+1)) are only sketched. In the revised manuscript we will expand this section with a dedicated lemma that (i) exhibits a concrete t for which min_i ||t v_i|| equals the claimed interior value, (ii) proves no larger min is possible by exhaustive case analysis on the possible nearest-integer configurations modulo 1, and (iii) shows the value lies strictly between consecutive terms of the sequence s/(4s+1) by direct fractional-part comparison. These additions will make the counterexamples fully rigorous without altering the overall claims. revision: yes

Circularity Check

0 steps flagged

No circularity: disproof via explicit constructions and direct ML computation

full rationale

The paper's central claim is a disproof of the Loneliness Spectrum Conjecture for n=4 via an explicit infinite family of 4-tuples, with ML computed directly from the definition max_t min_i ||t v_i||. No equation or step reduces the interior gap values to a fitted parameter, self-citation, or ansatz imported from prior work by the same authors. The amended conjecture is confirmed only for cases with a common factor >=3 using direct arguments. All load-bearing steps are self-contained against the definition of ML and do not invoke uniqueness theorems or renamings that collapse to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work uses only the standard definition of the nearest-integer distance function and integer speed tuples; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (1)
  • standard math The function ||x|| denotes the distance from x to the nearest integer and satisfies the usual properties of a metric on the circle.
    Invoked in the definition of ML(v1,...,vn).

pith-pipeline@v0.9.0 · 5738 in / 1296 out tokens · 29165 ms · 2026-05-24T08:22:07.261590+00:00 · methodology

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