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Amending the Lonely Runner Spectrum Conjecture

Ho Tin Fan, Alec Sun

TL;DR

This work challenges Kravitz's Loneliness Spectrum Conjecture by producing an infinite family of counterexamples for $n=4$ and introducing an Amended Loneliness Spectrum Conjecture that allows a $k$-parameter in the form $ML=\frac{s}{ns+k}$. It develops modular/pre-jump techniques and a Shifted Lonely Runner framework to establish lower bounds such as $ML\ge \frac{1}{4}$ in broad gcd configurations, and it identifies the exceptional family $(1,2,3,12k)$ with $ML=\frac{3k}{12k+1}$, while validating the amended conjecture in many regimes (including cases where a pair has gcd $>3$). The authors complement their theoretical results with computational evidence suggesting the discrete spectrum persists under the amendment and propose a stronger conjecture $k\le n/2$ for discrete values, outlining future work to complete the $n=4$ proof and extend experiments. Overall, the paper reframes the spectrum of loneliness values, provides a concrete infinite counterexample family, and offers a robust toolkit for gcd-structured analyses in the Lonely Runner setting.

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 $δ$, 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.

Amending the Lonely Runner Spectrum Conjecture

TL;DR

This work challenges Kravitz's Loneliness Spectrum Conjecture by producing an infinite family of counterexamples for and introducing an Amended Loneliness Spectrum Conjecture that allows a -parameter in the form . It develops modular/pre-jump techniques and a Shifted Lonely Runner framework to establish lower bounds such as in broad gcd configurations, and it identifies the exceptional family with , while validating the amended conjecture in many regimes (including cases where a pair has gcd ). The authors complement their theoretical results with computational evidence suggesting the discrete spectrum persists under the amendment and propose a stronger conjecture for discrete values, outlining future work to complete the proof and extend experiments. Overall, the paper reframes the spectrum of loneliness values, provides a concrete infinite counterexample family, and offers a robust toolkit for gcd-structured analyses in the Lonely Runner setting.

Abstract

Let be the absolute distance from to the nearest integer. For a set of distinct positive integral speeds , we define its maximum loneliness, also known as the gap , to be The Loneliness Spectrum Conjecture, recently proposed by Kravitz (2021), asserts that We disprove the Loneliness Spectrum Conjecture for with an infinite family of counterexamples and propose an alternative conjecture. We confirm the amended conjecture for whenever there exists a pair of speeds with a common factor of at least and also prove some related results.
Paper Structure (14 sections, 19 theorems, 70 equations, 1 table)

This paper contains 14 sections, 19 theorems, 70 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. Very fast runners
  5. Achieving the exceptions
  6. Large common factors
  7. More techniques
  8. Proving three shifted runners
  9. Large common factor between two speeds
  10. Almost very lonely runners when $g = 3$
  11. Why is $(1,2,3,12k)$ an exception?
  12. Concluding remarks
  13. Addressing the discrete spectrum
  14. Future work

Key Result

Theorem 2.1

For $n \geq 4$, let $v_1, v_2, \ldots, v_n$ be positive integers with $\gcd(v_1,\ldots, v_n) = 1$ and $\gcd(v_1,\ldots,v_{n-1}) = g \geq 2$ such that the Lonely Runner Conjecture holds for $v_1, \ldots, v_{n-1}$. Then

Theorems & Definitions (41)

  • Conjecture 1.1
  • Conjecture 1.2: Loneliness Spectrum Conjecture kravitz
  • Conjecture 1.3: Amended Loneliness Spectrum Conjecture
  • Theorem 2.1
  • Theorem 2.2: See also Theorem \ref{['the0']}
  • Theorem 2.3: See also Theorem \ref{['the3']}, \ref{['the4']} and Proposition \ref{['prop1']}
  • Lemma 3.1: Bohman, Holzman, and Kleitman bohman
  • Corollary 3.1
  • proof
  • Lemma 3.2: Folklore; see Czerwiński--Grytczuk czerwinski2008, Kravitz kravitz
  • ...and 31 more