View-obstruction: A shorter proof for 6 lonely runners

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If x is a real number, we denote by 〈x〉 ∈ [0, 1) the fractional part of x: 〈x〉 = x - E(x), where E(x) is the integer part of x. We give a simple proof of the following version of the Lonely Runner Conjecture: if v1, . . . , v5 are positive integers, there exists a real number t such that 〈tvi〉 ∈ [1/6, 5/6] for each i in {1, . . . , 5}. Our proof requires a careful study of the different congruence classes modulo 6 of the speeds v1, . . . , v5, and is simply based on the consideration of some time t maximizing the distance of 〈tv1〉 to {0, 1} among the set of times t such that 〈tvt〉 ∈ [1/6, 5/6] for each i ≠ 1. In appendix, we also give elementary proofs, based on the same idea, for analogous versions of the conjecture with fewer integers. © 2004 Elsevier B.V. All rights reserved.




Renault, J. (2004). View-obstruction: A shorter proof for 6 lonely runners. Discrete Mathematics, 287(1–3), 93–101.

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