15 problems

Given pairwise distinct positive velocities $v_1,\dotsc,v_n \in \mathbb{R}$ and arbitrary starting points $s_1,\dotsc,s_n \in \mathbb{R}$, the distance from $s_j+tv_j$ to the neare…

The strong Lonely Runner Conjecture. For every integer $d\geq1$, there exists an integer $N\geq1$ such that the equivalent bounded-time and covering properties hold; in particular,

Universal denominator conjecture. There exists a constant $D$ such that, for any integer $d\geq D$, every non-tight speed tuple $\mathbf{v}\in\mathbb{Z}_{>0}^k$ with…

Cusick–Wills conjecture. For each $i\in\{1,\ldots,k+1\}$, there is a real number $t$ such that

Let $u_1,\ldots,u_k$ be non-zero integers, and let $\lVert x\rVert$ denote the distance from $x$ to the closest integer. Wills's conjecture. There is a real number $t$ such that ……

Let $V=\{v_1,v_2,\dots,v_n\}$ be a set of $n$ distinct positive integers, and define the maximum loneliness by … where $\mathbf{T}=\mathbf{R}/\mathbf{Z}$ and…

Let $n$ runners with distinct constant speeds run around the unit circle, starting at a common time and place. Identify the unit circle with $\mathbf{T}=\mathbf{R}/\mathbf{Z}$, and…

Let $Z_{\mathbf v}$ be a Strong Lonely Runner Zonotope, meaning a Lonely Runner Zonotope associated with a velocity vector $\mathbf v=(v_1,\dotsc,v_n)\in\mathbb{Z}_{>0}^n$ whose en…

Let $Z=\sum_{i=1}^n[\mathbf{0},\mathbf{u}_i]\subseteq\mathbb{R}^{n-1}$ be a lattice zonotope of dimension $n-1$ with $n$ generators, whose every $n-1$ generators are linearly indep…

Dyadic-denominator conjecture. For every $\mathbf{n}$ with $\gcd(\mathbf{n})=1$, there is always a suitable time of the form

The free-starting-positions obstruction conjecture. Every such affine line $\mathcal L$ intersects infinitely many of these hypercubes. This is an obstruction-theoretic equivalent…

For each positive integer $k$, let $c_k$ denote the constant in the proposed lower bound for the $k$-wise join probabilities of the loneliness events. Asymptotic conjecture for…

Let $X$ be the random variable used in the paper to count the relevant loneliness events for a set of distinct speeds $v_1,\dots,v_n$, and let $\delta<1$. Second-moment lower-bound…

Let $n\geq 1$ and let $v_1,\dots,v_n$ be distinct speeds. For $x\in\mathbb{R}$, write $\|x\|$ for the distance from $x$ to the nearest integer. Weak Lonely Runner Conjecture. There…

Wills's conjecture. If $1\le m_1<m_2<\ldots<m_k$, then