Four runners that are barely lonely
Abstract
For positive integers v1,…,vn let ML(v1,…,vn)=maxtmini|vit|, where |x| is the distance to the nearest integer. Kravitz asked which values below 1/n occur. For n=4, Jain and Kravitz showed that the values below 1/4 are m/(4m+2), m≥2, up to finitely many undetermined exceptions. We settle the four-runner case completely. A primitive quadruple has ML<1/4 if and only if it is, up to order, {1,2,3,4k}, or {x,y,x+y,x+2y} with gcd(x,y)=1 and ynot≡ x,2xpmod 4, or {1,3,4,14}. The values below 1/4 are exactly m/(4m+2) with m≥2, m≠3, so 3/14 is the only exception, and Conjecture 4.1 of Fan and Sun holds. A density lemma bounds the length of a quadruple on neither critical subtorus by $512$, and an exact search over 800,816,099 quadruples completes the proof. The same method shows that the two-dimensional subtori of (ℝ/ℤ)4 with loneliness below 2/5 take exactly the three-runner values in that range, which between 1/3 and 2/5 are the fractions p/q with 1≤ 2q−5p≤ 4. Hence the accumulation points of the four-runner spectrum in (0,2/5] are the three-runner values there, as Giri and Kravitz conjectured, and 9/25 is not the loneliness of a two-dimensional subtorus. For every n, assuming the conjecture for n−1 and n−2 speeds, we prove ML(v)>1/n−(n(n−1))n−2/(2|v|), which is sharp in |v|, and that every n-tuple with ML<1/n and |v|≥ n(n(n−1))2n−4 lies on a two-dimensional subtorus of loneliness 1/n. This answers a question of Cordella. For five speeds, a two-dimensional subtorus of (ℝ/ℤ)5 with loneliness below 2/9 is the product of a coordinate circle with the circle of {1,2,3,4} or {1,3,4,7}. We also give infinitely many subtori whose maximizers lie off every circle with a repeated speed.
Related
BibTeX
@misc{yi2026four,
title = {{Four runners that are barely lonely}},
author = {Hyunjun Yi and Karam Gill and Bryan Bu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40694},
eprint = {2610.40694},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40694},
}
RIS
TY - UNPB
TI - Four runners that are barely lonely
AU - Hyunjun Yi
AU - Karam Gill
AU - Bryan Bu
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.40694
UR - https://arxiv.org/abs/2610.40694
AB - For positive integers v_1,…,v_n let ML(v_1,…,v_n)=max_tmin_i|v_it|, where |x| is the distance to the nearest integer. Kravitz asked which values below 1/n occur. For n=4, Jain and Kravitz showed that the values below 1/4 are m/(4m+2), m≥2, up to finitely many undetermined exceptions. We settle the four-runner case completely. A primitive quadruple has ML<1/4 if and only if it is, up to order, {1,2,3,4k}, or {x,y,x+y,x+2y} with gcd(x,y)=1 and ynot≡ x,2xpmod 4, or {1,3,4,14}. The values below 1/4 are exactly m/(4m+2) with m≥2, m≠3, so 3/14 is the only exception, and Conjecture 4.1 of Fan and Sun holds. A density lemma bounds the length of a quadruple on neither critical subtorus by $512$, and an exact search over 800,816,099 quadruples completes the proof. The same method shows that the two-dimensional subtori of (ℝ/ℤ)^4 with loneliness below 2/5 take exactly the three-runner values in that range, which between 1/3 and 2/5 are the fractions p/q with 1≤ 2q−5p≤ 4. Hence the accumulation points of the four-runner spectrum in (0,2/5] are the three-runner values there, as Giri and Kravitz conjectured, and 9/25 is not the loneliness of a two-dimensional subtorus. For every n, assuming the conjecture for n−1 and n−2 speeds, we prove ML(v)>1/n−(n(n−1))^n−2/(2|v|), which is sharp in |v|, and that every n-tuple with ML<1/n and |v|≥ n(n(n−1))^2n−4 lies on a two-dimensional subtorus of loneliness 1/n. This answers a question of Cordella. For five speeds, a two-dimensional subtorus of (ℝ/ℤ)^5 with loneliness below 2/9 is the product of a coordinate circle with the circle of {1,2,3,4} or {1,3,4,7}. We also give infinitely many subtori whose maximizers lie off every circle with a repeated speed.
ER -