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This visual formulation of the conjecture was first published in 1998.{{sfn|Bienia|Goddyn|Gvozdjak|Sebő|1998|p=3}} In many formulations, including the original by Jörg M. Wills,{{Sfnm|1a1=Wills|1y=1967|2a1=Bienia|2a2=Goddyn|2a3=Gvozdjak|2a4=Sebő|2y=1998}}{{sfn|Wills|1967}} some simplifications are made. The runner to be lonely is stationary at 0 (with zero speed), and therefore <math>n-1</math> other runners, with nonzero speeds, are considered.{{efn|Some authors use the convention that <math>n</math> is the number of non-stationary runners, and thus the conjecture is that the gap of loneliness is at most <math>1/(n+1)</math>.{{sfn|Tao|2018}} }} The moving runners may be further restricted to ''positive'' speeds only: by symmetry, runners with speeds <math>x</math> and <math>-x</math> have the same distance from 0 at all times. Proving the result for any stationary runner implies the general result for all runners, since she can be made stationary by subtracting her speed from all runners, leaving her with zero speed. The conjecture then states that, for any collection <math>v_1,v_2,...,v_{n-1}</math> of positive, distinct speeds, there exists some time <math>t>0</math> such that
<math display="block">\frac{1}{n}\leq \operatorname{frac}(v_it)\leq 1-\frac{1}{n}\qquad (i=1,...,n-1),</math>
where <math>\operatorname{frac}(x)</math> denotes the [[fractional part]] of <math>x</math>.{{sfn|Bohman|Holzman|Kleitman|2001|p=2}} Interpreted visually, if the runners are running counterclockwise, the middle term of the inequality is the distance from the origin to the <math>i</math>th runner at time <math>t</math>, measured counterclockwise.{{Efn|For example, if the origin is at a 6 o'clock position, a runner at the 9 o'clock position will have <math>\operatorname{frac}(vt)=3/4</math>.}} Wills' conjecture was part of his work in [[Diophantine approximation]],{{sfnm|1a1=Wills|1y=1967|2a1=Betke|2a2=Wills|2y=1972}} the study of how closely fractions can approximate irrational numbers. His conjecture concerns how well multiple real numbers
== Implications ==
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