The lonely runner conjecture asserts that, for any 
distinct positive integers 
,
there is a real number 
such that
 |
where 
denotes distance to the nearest integer, not a vector
norm. In the running interpretation, a stationary runner and 
other runners move on a unit-circumference circle,
and every runner should at some time be at distance at least 
from all the others (Hu and Jing 2026).
Hu and Jing (2026) proved that there is an absolute constant 
such that, for every 
and every such speed set,
 |
This is a fixed proportional improvement over 
, not the full conjectured bound. Hu and Jing (2026) describe
ChatGPT assistance with checking and making arguments rigorous and with refinements,
while attributing the strategy and framework to the authors.
