Lonely Runner Relations
Abstract
We study the Lonely Runner Conjecture (LRC), conceived by Jörg M. Wills in the 1960's: Given positive integers n1, n2, …, nk, there exists a positive real number t such that for all 1 ≤ j ≤ k the distance of t nj to the nearest integer is at least ( 1 )/( k+1 ). We prove that for any counterexample or tight instance n of LRC, m · n = 0 for some m ∈ ℤk with 0 < | m |1 ≤ min(2k+3, ( k+1 )/( k−1 ) flt(k)) where flt(k) denotes Khinchin's (1948) flatness constant limiting the lattice width of a k-dimensional convex body without interior integer points. In other words, potential counterexamples to LRC lie on a finite set of hyperplanes in the parameter space. Our proofs use Fourier analysis and a geometric reformulation of LRC, and our results generalize to the situation of shifted lonely runners of varying measures of loneliness. Our results imply and generalize a theorem of Czerwiński (2012) that when we choose n at random then, with probability tending to 1, the measure of loneliness 1/( k+1 ) can be replaced by 1/2 − ε.
Related
BibTeX
@misc{beck2026lonely,
title = {{Lonely Runner Relations}},
author = {Matthias Beck and Samuel Everett},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06259},
eprint = {2609.06259},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06259},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lonely Runner Relations
AU - Matthias Beck
AU - Samuel Everett
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.06259
UR - https://arxiv.org/abs/2609.06259
AB - We study the Lonely Runner Conjecture (LRC), conceived by Jörg M. Wills in the 1960's: Given positive integers n_1, n_2, …, n_k, there exists a positive real number t such that for all 1 ≤ j ≤ k the distance of t n_j to the nearest integer is at least ( 1 )/( k+1 ). We prove that for any counterexample or tight instance n of LRC, m · n = 0 for some m ∈ ℤ^k with 0 < | m |_1 ≤ min(2k+3, ( k+1 )/( k−1 ) flt(k)) where flt(k) denotes Khinchin's (1948) flatness constant limiting the lattice width of a k-dimensional convex body without interior integer points. In other words, potential counterexamples to LRC lie on a finite set of hyperplanes in the parameter space. Our proofs use Fourier analysis and a geometric reformulation of LRC, and our results generalize to the situation of shifted lonely runners of varying measures of loneliness. Our results imply and generalize a theorem of Czerwiński (2012) that when we choose n at random then, with probability tending to 1, the measure of loneliness 1/( k+1 ) can be replaced by 1/2 − ε.
ER -