18 problems

Lonely runner conjecture. There exists $x\in\mathbb{T}$ such that

Let $v_1,\ldots,v_n$ be positive integers, and let … where $\lVert x\rVert$ denotes the absolute distance from $x$ to the nearest integer. The Loneliness Spectrum Conjecture. One h…

Function-field Lonely Runner Conjecture. For every prime power $q$ and every finite set $F\subset\mathbb{F}_q[X]\setminus\{0\}$ of polynomials satisfying

Let $V$ be an $n$-set of positive integer speeds, let $v_n$ denote the largest speed, and let $\|x\|$ be the distance from $x$ to the nearest integer. Conjecture on lonely times. F…

Timely Lonely Runner Conjecture. For every $n\in\mathbb{N}$ there is $N$ such that, for every $n$-set $V$ of positive speeds,

For a positive-integer speed set $V$, let $\kappa(V)$ denote its gap of loneliness, and let $n=|V|$. Amended Loneliness Spectrum Conjecture. For every $n\in\mathbb{N}$ and every…

Let $V=\{v_1,\dots,v_n\}$ be an $n$-set of pairwise distinct real speeds, and let $\|x\|$ denote the distance from $x$ to the nearest integer. Single Lonely Runner Conjecture. For…

Let $v_1,v_2,v_3,v_4$ be positive integers, and let … where $\lVert x\rVert$ denotes the absolute distance from $x$ to the nearest integer. The restricted amended Loneliness Spectr…

Let $v_1,\ldots,v_n$ be positive integers, and let … where $\lVert x\rVert$ denotes the absolute distance from $x$ to the nearest integer. The amended Loneliness Spectrum Conjectur…

Short-progression exclusion conjecture. One has

Weak Loneliness Spectrum Conjecture. For every integer $n\geq4$, there exists a function $f_n:\mathbb{N}\to\mathbb{N}$ such that, whenever $v_n>f_n(v_{n-1})$, either

For $t\in\mathbf{R}/\mathbf{Z}$, let $\|t\|_{\mathbf{R}/\mathbf{Z}}$ be the distance from any representative of $t$ to the nearest integer. For an $n$-tuple of non-zero integers…

Haralambis' equality conjecture. Every three-element $D$-set satisfies

Let $n$ be a positive integer. Let $Z\subseteq\mathbb{R}^n$ be a zonotope generated by $n+1$ vectors of $\mathbb{Z}^n$ in LGP (general linear position), and let $\boldsymbol{x}$ be…

The finite-group formulation. There exist a natural number $n$ and an element $x\in\mathbb{Z}_n$ such that

The $K_n'$ formulation.

The equality formulation.

The original formulation.