11 problems

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 $A$ 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 $({\mathcal A},{\mathcal B})$ 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 $\Omega=A\times B$. Product spectrality conjecture. The product $\Omega$ is spectral…

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…

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

Let $\Gamma\subset\mathbb R$ be a set. A spectrum for a spectral set $\Omega$ is a set of frequencies whose normalized exponentials form an orthonormal basis of $L^2(\Omega)$. A pr…

Finite spectral-set conjecture. The following conditions are equivalent:

Universal spectrum conjecture. Under these assumptions, $\Gamma$ has a universal spectrum of the form