Mixed thresholds in the Lonely Runner Conjecture

Abstract

The Lonely Runner Conjecture states that if $k+1$ runners start at the same point on a unit-length circular track and run with distinct constant speeds, then each runner is at some time at least $1/(k+1)$-distant from every other runner. Equivalently, for every tuple of $k$ distinct positive integer speeds $s_1,\ldots,s_k$, there is a real number $t$ such that $\|s_i t\|\geq \frac{1}{k+1}$ for all $i$. We introduce and study a version of the conjecture in which the required distances may vary with $i$. For $\mathbf d=(d_1,\ldots,d_k)\in(0,1/2]^k$, let $\mathsf{MLPS}_k$ be the set of vectors such that, for every choice of distinct positive integer speeds $s_1,\ldots,s_k$, there is a real number $t$ with $\|s_i t\|\geq d_i$ for all $i$. We give an exact characterization of $\mathsf{MLPS}_2$. We also use Fourier series for distance-threshold indicator functions to obtain an arithmetic progression summation formula and an exact two-function integral formula for unequal thresholds.