Tight instances of the Lonely Runner Conjecture: one-entry modifications and the growth of speeds
Abstract
For a set V of n−1 distinct positive integers let LR(V)=maxtminv∈ V|vt|, where |x| is the distance from x to the nearest integer. The Lonely Runner Conjecture asserts LR(V)≥ 1/n, and V is called tight if equality holds. Goddyn and Wong determined which of the sets obtained from [n−1]={1,…,n−1} by multiplying a single entry by an integer are tight. We complete the classification of arbitrary one-entry modifications ([n−1]∖{r})∪{w}, w≥ n: for n≥4 such a set is tight if and only if it is of Goddyn--Wong type or is one of the two sporadic sets {1,3,4,7} and {1,3,4,5,9} found by Wills. The proof is elementary and rests on an exact description of the region left uncovered by the retained speeds. We then show that the largest speed M(n) occurring in a tight one-entry modification satisfies M(n)<0.60 nlog n+52 n and limsupn→∞M(n)/(nlog n)=1/2; in particular, tight sets are not confined to speeds O(n). Along the way we give a short proof of Liu's closed formula for LR([n−1]∖{2}) and show that LR([n−1]∖{r})>1/(n−1) for every r≤ n−2−2√n, extending the range r≤(n−1)/2 covered by Liu's theorem. For two-entry modifications we show that no tight instance removes a speed q with 2≤ q≤ n/10 once n≥max(40,10q), whatever the second removed speed and the inserted speeds are; an exact-arithmetic census for 8≤ n≤140 with inserted speeds at most 8n finds only two tight two-entry modifications, both already listed by Goddyn and Wong.
Related
BibTeX
@misc{zhang2026tight,
title = {{Tight instances of the Lonely Runner Conjecture: one-entry modifications and the growth of speeds}},
author = {Yuhan Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.13599},
eprint = {2608.13599},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.13599},
}
RIS
TY - UNPB
TI - Tight instances of the Lonely Runner Conjecture: one-entry modifications and the growth of speeds
AU - Yuhan Zhang
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2608.13599
UR - https://arxiv.org/abs/2608.13599
AB - For a set V of n−1 distinct positive integers let LR(V)=max_tmin_v∈ V|vt|, where |x| is the distance from x to the nearest integer. The Lonely Runner Conjecture asserts LR(V)≥ 1/n, and V is called tight if equality holds. Goddyn and Wong determined which of the sets obtained from [n−1]={1,…,n−1} by multiplying a single entry by an integer are tight. We complete the classification of arbitrary one-entry modifications ([n−1]∖{r})∪{w}, w≥ n: for n≥4 such a set is tight if and only if it is of Goddyn--Wong type or is one of the two sporadic sets {1,3,4,7} and {1,3,4,5,9} found by Wills. The proof is elementary and rests on an exact description of the region left uncovered by the retained speeds. We then show that the largest speed M(n) occurring in a tight one-entry modification satisfies M(n)<0.60 nlog n+52 n and limsup_n→∞M(n)/(nlog n)=1/2; in particular, tight sets are not confined to speeds O(n). Along the way we give a short proof of Liu's closed formula for LR([n−1]∖{2}) and show that LR([n−1]∖{r})>1/(n−1) for every r≤ n−2−2√n, extending the range r≤(n−1)/2 covered by Liu's theorem. For two-entry modifications we show that no tight instance removes a speed q with 2≤ q≤ n/10 once n≥max(40,10q), whatever the second removed speed and the inserted speeds are; an exact-arithmetic census for 8≤ n≤140 with inserted speeds at most 8n finds only two tight two-entry modifications, both already listed by Goddyn and Wong.
ER -