15 problems

A measurable set $E\subset\mathbb{R}^{d}$ of positive Lebesgue measure is spectral if there exists a countable set $\Lambda\subset\mathbb{R}^{d}$ such that the exponential family ……

Let $T$ be a tile in $\mathbb{Z}^n$. A tiling by translated copies of $T$ is periodic if it is invariant under a nonzero translation of $\mathbb{Z}^n$. Periodic tiling conjecture.…

Let $\mathbf{G}(g,T,S)$ be a Gabor system that is an orthonormal basis in $L^2(\mathbb{R}^d)$, with window function $g$ of compact support. A measurable set…

Let $A(x)$ be a polynomial whose coefficients are nonnegative integers. Write $A(x)=\sum_{a\in A}x^a$ when a finite set $A\subset\\{0,1,2,\dots\\}$ supplies the coefficients, and s…

Let ${\cal T}={\mathbb Z}+\frac{1}{m}A$, where $A\subseteq{\mathbb Z}$ is reduced modulo $m$ and admits a factorization $A\oplus B={\mathbb Z}/m{\mathbb Z}$. A universal spectrum f…

Let $E \subset \mathbb{R}^n$ and $F \subset \mathbb{R}^m$ be bounded and measurable sets, and set $\Omega=E\times F$. A set is spectral if its $L^2$ space admits an orthogonal basi…

A rectangle tiling set is a finite set $R$ of rectangles, and a simply connected region is a region in the plane without holes that is to be tiled using translated copies of rectan…

Let $R$ be an $n\times n$ expanding integer matrix and let $0\in\mathcal D\subset\mathbb Z^n$ satisfy $\#\mathcal D=|\det(R)|$. Define $T(R,\mathcal D)$ to be the associated self-a…

Let $p$ be a prime and $d\geq 1$. A set $E\subseteq (\mathbb Z/p\mathbb Z)^d$ is tiling if there exists $T\subseteq (\mathbb Z/p\mathbb Z)^d$ such that the translates…

Let $p$ be a prime, and let $E,T\subseteq (\mathbb Z/p\mathbb Z)^3$ be sets such that the translates $\{E+t\mid t\in T\}$ partition $(\mathbb Z/p\mathbb Z)^3$. Thus $(E,T)$ is a fa…

Conjecture for non-prismatic convex polytopes. A corresponding version of Theorem should hold for $A$ and $H(A)$.

Let $g$ be a compactly supported window function and let $\Lambda\subseteq\mathbb{R}^{2d}$ be a separable time-frequency set, so that $\Lambda=\mathcal{J}\times\Gamma$ for discrete…

Spectral set conjecture. The set $\Omega$ is spectral if and only if it tiles $G$ by translation.

Let $M$ be a standard multigraph on $n$ vertices, where $n$ is divisible by $3$. Let $\delta(M)$ denote its minimum degree, let $H_M$ be its heavy-edge graph, and let a $T_5$-facto…

Konyagin–Łaba conjecture. The set $A$ tiles the integers by translation if and only if both $(T_1)$ and $(T_2)$ hold.