Odd denominators in the Lonely Runner spectrum for six speeds

Source abstract

For distinct positive integers v_1, ..., v_n let ML(v_1, ..., v_n) be the largest number L such that at some time t every t_vi is at distance at least L from the nearest integer; the Lonely Runner Conjecture asserts that ML >= 1/(n+1). Write ML = p/q in lowest terms. Kravitz conjectured that whenever ML < 1/n one has q = np + 1; Fan and Sun found counterexamples for n = 4, conjectured that q = np + k with 1 <= k <= n always holds, and observed that in their data for n = 6 only k = 1 and k = 3 occur. We explain this observation. For six speeds we show that all but finitely many tuples with ML < 1/6 satisfy ML = (P-1)/(6P) for an integer P congruent to 1 or 5 modulo 6; in particular k is 1 or 3 and the denominator q is odd. The proof determines the three infinite two-parameter families of tuples on which such values concentrate, computes ML exactly on each family, and describes exactly where k = 3 occurs. An exhaustive search over the 2 x 10^9 sextuples with speeds at most 110 finds no exception. For five speeds the same method, together with Chen's classification of the tuples attaining ML = 1/5, shows that all but finitely many tuples with ML < 1/5 satisfy Kravitz's original conjecture. The computations are exact and the code is provided.