Maximum-loneliness formulation of the Lonely Runner Conjecture

Let V={v1,v2,,vn}V=\{v_1,v_2,\dots,v_n\} be a set of nn distinct positive integers, and define the maximum loneliness by

ML(V)=ML(v1,,vn):=maxtT(minj=1,,ntvjT),\operatorname{ML}(V)=\operatorname{ML}(v_1,\dots,v_n)\vcentcolon=\max_{t\in\mathbf{T}}\left(\min_{j=1,\dots,n}\lVert tv_j\rVert_{\mathbf{T}}\right),

where T=R/Z\mathbf{T}=\mathbf{R}/\mathbf{Z} and T\lVert\cdot\rVert_{\mathbf{T}} denotes the distance to the nearest integer. Lonely Runner Conjecture, maximum-loneliness formulation. If v1,v2,,vnv_1,v_2,\dots,v_n are nn distinct positive integers, then

ML(v1,,vn)1n+1.\operatorname{ML}(v_1,\dots,v_n)\geqslant \frac{1}{n+1}.

This is the succinct formulation obtained after translating one runner's speed to zero; it is equivalent to the Lonely Runner Conjecture for n+1n+1 runners. The notation is attributed in the source to Kravitz, and the conjecture remains open.

Sources & referencesView supporting material

Primary source

Benjamin Bedert, “Riesz products and the Lonely Runner Conjecture: A wider gap of loneliness”, arXiv:2511.16636 (2025).

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