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- W4381552204 abstract "Let $|x|$ be the absolute distance from $x$ to the nearest integer. For a set of distinct positive integral speeds $v_1, ldots, v_n$, we define its maximum loneliness to be $$text{ML}(v_1,ldots,v_n) = max_{t in mathbb{R}}min_{1 leq i leq n} |tv_i|$$ The Loneliness Spectrum Conjecture, recently proposed by Kravitz, asserts that $$exists s in mathbb{N}, text{ML}(v_1,ldots,v_n) = frac{s} {sn + 1} text{ or } text{ML}(v_1,ldots,v_n) ge frac{1}{n}$$ We disprove the Loneliness Spectrum Conjecture for $n = 4$ and propose an alternative conjecture. We confirm the amended conjecture for $n = 4$ if any pair of speeds share a common factor of at least $3$ and also prove some related results." @default.
- W4381552204 created "2023-06-22" @default.
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- W4381552204 date "2023-06-17" @default.
- W4381552204 modified "2023-10-18" @default.
- W4381552204 title "Amending the Lonely Runner Spectrum Conjecture" @default.
- W4381552204 doi "https://doi.org/10.48550/arxiv.2306.10417" @default.
- W4381552204 hasPublicationYear "2023" @default.
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