Tight instances of the Lonely Runner Conjecture: complete classification of one-entry modifications, a new infinite family, and the growth bound

Abstract

For a set V of n-1 distinct positive integers write LR(V) = max_t min_{v in V} ||vt||, where ||x|| is the distance from x to the nearest integer; V is tight if LR(V) = 1/n, the value predicted by the Lonely Runner Conjecture. The baseline [n-1] = {1,...,n-1} is tight for every n, and Perarnau and Serra list the characterization of tight instances as Problem 1 of their survey, noting no further progress since Goddyn and Wong, who classified the multiple case and proved finiteness for each fixed deleted speed. We settle the one-entry case completely: ([n-1] minus {r}) union {w} with w > n-1 is tight if and only if either 2r > n-1, r divides w and the Goddyn-Wong gcd criterion holds, or (n,r,w) = (5,2,7) or (6,2,9). In particular no tight one-entry modification exists for 3 <= r <= (n-1)/2, and the only tight cases beyond the Goddyn-Wong multiples are the two sporadic sets of Wills, {1,3,4,7} and {1,3,4,5,9}. The proof is purely theoretical: the connected components of the uncovered region U(n,r) are determined exactly in both regimes 2r > n-1 and 2r <= n-1, yielding the effective bound w <= 4rI/(2s-I), with s = n-r and I the least integer of [s, n-1] coprime to r; a single inequality, proved via the Jacobsthal function, closes the mid-range without computation. Tight one-entry modifications also satisfy max V <= 0.60 n log n + 52 n unconditionally, with sharp leading constant 1/2 along n = p#+2 (p# the primorial), so no linear bound confines tight instances. We further isolate an explicit CRT doubling subfamily of the Goddyn-Wong multi-acceleration theorem and prove for two-speed replacements that no tight instance contains a removed speed q with 2 <= q <= n/10 when n >= max(40,10q), whatever the second removal and the inserted speeds. An exact census over all n <= 140, w <= 8n finds only two tight two-swaps, the Wills set {1,4,5,6,7,11,13} and the Goddyn-Wong doubling at n = 74.