28 problems

Let $A$ be an $n\times n$ matrix, let $0<|q|\leq 1$, and let $\Omega_q$ denote the scaled $q$-numerical range introduced in the source. For a polynomial $p$ with complex coefficien…

Let $A \subset \mathbb{R}^n$ and $B \subset \mathbb{R}^m$ be bounded, measurable sets, and let $\Omega=A\times B$. Product spectrality conjecture. The product $\Omega$ is spectral…

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 $\mathcal T$ be a tiling set, and call $\Lambda_{\mathcal T}$ a universal spectrum if every set $\Omega$ that tiles $\mathbb R^d$ by $\mathcal T$ is spectral with spectrum…

Let $I^{d}$ denote the $d$-dimensional unit cube, and let $L\subset\mathbb{R}^{d}$. The cube spectral-tiling conjecture. $(I^{d},L)$ is a spectral pair if and only if $(I^{d},L)$ i…

Let $L\subset\mathbb{R}^{d}$. A spectrum is a set $L$ for which there exists a set $\Omega$ such that $(\Omega,L)$ is a spectral pair, and a tiling set is a set $L$ for which there…

Let $G$ be a finite Abelian group and $T\subset G$ a tile. Suppose that $S\subset\widehat G$ satisfies … where $Z_T$ is the Fourier zero-set of $T$. Such an $S$ is a universal spec…

Let $G$ be a finite Abelian group, and let $T\subset G$ be a tile. A subset $S\subset\widehat G$ is a universal spectrum of $T$ if $S$ is a spectrum of every tiling complement $T'$…

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 $E \subset \mathbb{R}^n$ 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…

Let $({\mathcal A},{\mathcal B})$ be a distributed spectral pair in $\{0,1,\ldots,M-1\}$. A distributed spectral pair classification conjecture. The pair…

Let $A\subset\mathbb{R}^n$ and $B\subset\mathbb{R}^m$ be bounded, measurable sets, and let … A measurable set is spectral if its indicator function admits an orthogonal basis of ex…

Let $D=\{x\in\mathbf{R}^2\colon \|x\|<1\}$ be the unit disk, and let $\Lambda\subset\mathbf{R}^2$ be a frequency set. Define … Assume that $E(\Lambda)$ is an orthogonal system in…

Let $\mathcal A$ be an integer alphabet, let $X_{\mathcal A}(\beta)$ denote the associated spectrum, and consider Pisot numbers $\beta_n$ of increasing degree in an interval…

Let $p$ be a prime, and let $n,d\geq 1$. Consider the finite abelian group $\mathbb{Z}_{p^n}^d$. The p-group tiling conjecture. Every tile in $\mathbb{Z}_{p^n}^d$ is a spectral set…

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 $D=\\{x\in\mathbb{R}^{2}:\lVert x\rVert\leq 1\\}$ be the unit disc. A measurable set is a Riesz set if its $L^{2}$ space admits a Riesz basis of exponentials. Olevskii–Ulanovsk…

Let $H$ be a complex Hilbert space, let $T\in\mathcal{B}(H)$, and let $W(T)=\{\langle Tx,x\rangle:\|x\|=1\}$ be its numerical range. A set $\Omega$ is a complete $K$-spectral set f…

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…

Let $R$ be a $d\times d$ expansive integer matrix, and let $B\subset\mathbb{Z}^d$ contain $0$, with the elements of $B$ mutually incongruent modulo $R$. Suppose there are a set…

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

Bounded-doubling spectrum conjecture. There exists a subset $A'\subseteq A$ with $|A'|\geq |A|/C$ such that

Foiaș–Williams conjecture. The closure of the numerical range is a complete $K$-spectral set for $A$.

Universal Tiling Conjecture $\operatorname{UTC}(p)$. There exists a common tiling set $\mathcal T\subset\mathbb{Z}$ such that each $A_i$ tiles $\mathbb{Z}$ by translations from…