35 problems

Let $d$ be a positive integer and let $\mathbb{Z}^{\Box d}$ denote the $d$-dimensional integer lattice with the local connectivity represented by the Cartesian product. A firefight…

Prime divisibility embedding conjecture. There is a Euclidean motion $T$ such that

Divisibility embedding conjecture. There is a set $N\subset\mathbb{Z}^n$ such that $k\cdot N$ is congruent to $M$.

Embedding conjecture. One can find a Euclidean motion $T$ such that

Let $\mathbb{Z}^n$ have the Manhattan metric, let $\{0,1\}^n\subseteq\mathbb{Z}^n$ have the induced metric, and let $\widetilde{H}_i(-;\mathbb{Z})$ denote reduced homology with int…

Let $\mathbb{Z}^n$ carry the Manhattan metric, also called the standard word metric, and let $\mathcal{VR}(\mathbb{Z}^n;r)$ denote its Vietoris–Rips complex at scale $r$. Zaremsky'…

Abundance conjecture. In analogy with the authors' results in the continuous setting, there are plenty of independent doubling minimizers in $\mathbb{Z}^d$ for every $d\geq 3$.

Let an icube in $\mathbb{Z}^8$ be a set of pairwise orthogonal vectors of equal norm. Extension conjecture. Any icube in $\mathbb{Z}^8$ can be extended to an $8$-icube. This is mot…

Let $n$ be a positive integer, and let a monotile be a single tile used by translations to tile subsets of the integer lattice $\mathbb{Z}^n$. Undecidability conjecture. Translatio…

Norm-extension conjecture. Then $\lVert\cdot\rVert$ is the restriction to $G$ of a norm on $\mathbb Z^3$.

For positive integers $n$ and scales $r$, let $\mathrm{Rips}(\mathbb Z^n,r)$ denote the Rips complex of the integer lattice, with the metric understood from the paper. The paper di…

For each positive integer $n$, let $\mathrm{Rips}((\mathbb Z^n,d_1),r)$ denote the Rips complex of the $n$-dimensional integer lattice with its Manhattan metric at scale $r$. The p…

Let $S\subseteq\mathbb{N}^2$ contain no three points on a common line. No-three-in-line density conjecture. … Although finite $n\times n$ configurations have the elementary upper b…

Let $d\in\mathbb{N}$, let $M_1,M_2\in M_{d\times d}(\mathbb{Z})$, and let $A\subseteq\{1,\dots,N\}^d$. Finitary box recurrence conjecture. If $M_2-M_1$ is nonsingular, then for eve…

Let $\alpha,n,d\in\mathbb{Z}^{+}$, let $C\subset\mathbb{Z}^{d}$ satisfy $|C|=n$, and let $\mathcal{T}(\alpha,C)$ denote the set of canonical translational tilings of $C$ by tiles o…

Let $d\in\mathbb{Z}^{+}$, let $\alpha\in\mathbb{Z}^{+}$, and let $C\subset\mathbb{Z}^{d}$ be finite and contiguous. Let $C'\subset\mathbb{Z}^{d}$ be any finite set, and let…

Periodic co-tile conjecture. If there exists $A\subseteq\mathbb{Z}^d$ such that

Let $d\geq 1$ and $p\in(0,1)$. On the graph $\mathbbm{Z}^d$, whose vertices at Euclidean distance $1$ are joined by an edge, use the componentwise order $u\leq v$ defined by…

Let $F$ be a finite set of integer points in $\mathbb{Z}^d$, and let $S$ be a set of integer translations such that $F$ multi-tiles $\mathbb{Z}^d$ using translations from $S$. Peri…

Weakly periodic tiling conjecture. There exists a weakly periodic $F$-tiling of $\mathbb Z^d$; more precisely, there exists an $F$-tiling $T\subseteq\mathbb Z^d$ admitting such a f…

Periodic tiling conjecture. If there is an $F$-tiling of $\mathbb Z^n$ for some cluster $F$, then there is a fully periodic $F$-tiling.

For $d\geq 3$, let $p_{1,c}(\mathbb{Z}^d)$ denote the critical probability for $1$-independent percolation on the integer lattice. High-dimensional percolation conjecture. There ex…

Let $\operatorname{MD}_{\mathrm{coins}}$ denote the coin-valued median dynamics process on $\mathbb{Z}^d$, and let $\eta_t(\underline{0})$ be the opinion at the origin. Coin-median…

Let $\operatorname{MD}_{\mathrm{coins}}$ denote the coin-valued median dynamics process on $\mathbb{Z}^d$, and let $\eta_t(\underline{0})$ be the opinion at the origin. Coin-median…

Consider median dynamics on $\mathbb{Z}^2$, with $\eta_\infty(0)$ denoting the limiting opinion at the origin. Stable-structure tail conjecture. … The exponent $4$ comes from the s…