Tight instances of the Lonely Runner Conjecture: one-entry modifications and the growth of speeds

For a set $V$ of $n-1$ distinct positive integers let $\mathrm{L}\mathrm{R}(V)={\max }_t{\min }_{v\in V}\mathrm{\parallel }vt\mathrm{\parallel }$, where $\mathrm{\parallel }x\mathrm{\parallel }$ is the distance from $x$ to the nearest integer. The Lonely Runner Conjecture asserts $\mathrm{L}\mathrm{R}(V)\ge 1\mathrm{/}n$, and $V$ is called tight if equality holds. Goddyn and Wong determined which of the sets obtained from $[n-1]=\{1,\dots ,n-1\}$ by multiplying a single entry by an integer are tight. We complete the classification of arbitrary one-entry modifications $([n-1]\setminus \{r\})\cup \{w\}$, $w\ge n$: for $n\ge 4$ such a set is tight if and only if it is of Goddyn–Wong type or is one of the two sporadic sets $\{1,3,4,7\}$ and $\{1,3,4,5,9\}$ found by Wills. The proof is elementary and rests on an exact description of the region left uncovered by the retained speeds. We then show that the largest speed $M(n)$ occurring in a tight one-entry modification satisfies $M(n)<0.60n\log n+52n$ and ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}M(n)\mathrm{/}(n\log n)=1\mathrm{/}2$; in particular, tight sets are not confined to speeds $O(n)$. Along the way we give a short proof of Liu’s closed formula for $\mathrm{L}\mathrm{R}([n-1]\setminus \{2\})$ and show that $\mathrm{L}\mathrm{R}([n-1]\setminus \{r\})>1\mathrm{/}(n-1)$ for every $r\le n-2-2\sqrt{n}$, extending the range $r\le (n-1)\mathrm{/}2$ covered by Liu’s theorem. For two-entry modifications we show that no tight instance removes a speed $q$ with $2\le q\le n\mathrm{/}10$ once $n\ge \max (40,10q)$, whatever the second removed speed and the inserted speeds are; an exact-arithmetic census for $8\le n\le 140$ with inserted speeds at most $8n$ finds only two tight two-entry modifications, both already listed by Goddyn and Wong.