Four runners that are barely lonely

Abstract

For positive integers $v_1,\dots ,v_n$ let $\mathrm{ML}(v_1,\dots ,v_n)={\max }_t{\min }_i\Vert v_{i}t\Vert$, where $\Vert x\Vert$ is the distance to the nearest integer. Kravitz asked which values below $1/n$ occur. For $n=4$, Jain and Kravitz showed that the values below $1/4$ are $m/(4m+2)$, $m\ge 2$, up to finitely many undetermined exceptions. We settle the four-runner case completely. A primitive quadruple has $\mathrm{ML}<1/4$ if and only if it is, up to order, $\{1,2,3,4k\}$, or $\{x,y,x+y,x+2y\}$ with $\gcd (x,y)=1$ and $y\not\equiv x,2x(\mathrm{mod}4)$, or $\{1,3,4,14\}$. The values below $1/4$ are exactly $m/(4m+2)$ with $m\ge 2$, $m\ne 3$, so $3/14$ is the only exception, and Conjecture 4.1 of Fan and Sun holds. A density lemma bounds the length of a quadruple on neither critical subtorus by $512$, and an exact search over 800,816,099 quadruples completes the proof. The same method shows that the two-dimensional subtori of $(\mathbb{R}/\mathbb{Z}{)}^4$ with loneliness below $2/5$ take exactly the three-runner values in that range, which between $1/3$ and $2/5$ are the fractions $p/q$ with $1\le 2q-5p\le 4$. Hence the accumulation points of the four-runner spectrum in $(0,2/5]$ are the three-runner values there, as Giri and Kravitz conjectured, and $9/25$ is not the loneliness of a two-dimensional subtorus. For every $n$, assuming the conjecture for $n-1$ and $n-2$ speeds, we prove $\mathrm{ML}(v)>1/n-(n(n-1){)}^{n-2}/(2|v|)$, which is sharp in $|v|$, and that every $n$-tuple with $\mathrm{ML}<1/n$ and $|v|\ge n(n(n-1){)}^{2n-4}$ lies on a two-dimensional subtorus of loneliness $1/n$. This answers a question of Cordella. For five speeds, a two-dimensional subtorus of $(\mathbb{R}/\mathbb{Z}{)}^5$ with loneliness below $2/9$ is the product of a coordinate circle with the circle of $\{1,2,3,4\}$ or $\{1,3,4,7\}$. We also give infinitely many subtori whose maximizers lie off every circle with a repeated speed.

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