Amending the Lonely Runner Spectrum Conjecture - Fan, Ho Tin; Sun, Alec (The Electronic Journal of Combinatorics, 2026)
Let x Vert be the absolute distance from x to the nearest integer. For a set of distinct positive integer speeds v_1, …, v_n, we define its maximum loneliness, also known as the gap δ, to be ML(v_1,…,v_n) = δ(v_1,…,v_n) = _t R _1 ≤ i ≤ n tv_i Vert. The Loneliness Spectrum Conjecture, recently proposed by Kravitz, asserts that s N, ML(v_1,…,v_n) = s sn + 1 or ML(v_1,…,v_n) ≥ 1n. We disprove the Loneliness Spectrum Conjecture for n = 4 with an infinite family of counterexamples and propose an alternative conjecture. We confirm the amended conjecture for n = 4 whenever there exists a pair of speeds with a common factor of at least 3 and prove some related results.