60 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 ……

Łaba–Wang conjecture. If the self-similar measure $\mu_{b,D}$ is a spectral measure, then $\alpha D$ is an integer tile for some $\alpha\in\mathbb{R}$.

Let $R\in M_n(\mathbb{Z})$ be an expanding matrix, and let $D,L\subset\mathbb{Z}^n$ be finite digit sets with $\#D=\#L$. The pair $(R,D)$ is admissible when $(R,D,L)$ is a Hadamard…

Tang–Kohmoto conjecture. An absolutely continuous spectrum is dominated by points with the trivial scaling index

Fuglede's spectral set conjecture. A measurable set … by translations.

Bauer–Golinelli conjecture. The following phase transition occurs: if $c<e$, then $\mu_c$ has no extended states at $0$; if $c>e$, then $\mu_c$ has extended states at $0$.

Let $R$ be an expansive $d$ by $d$ matrix over $\b7bZ$, and set $S=R^T$ to be the transposed matrix. Let $B,L\subset\mathbb{R}^d$ have the same cardinality $N$, and assume that the…

Let $X$ and $Y$ be colored graphs. Write $X*Y$ for the colored graph obtained by putting a copy of $X$ at each vertex of $Y$, let $\mu(Z)$ denote the spectral measure of the charac…

Let … is a spectral measure, meaning that $L^2$ has an orthogonal basis of complex exponentials. Let $=_j$ be the digit set. The spectral self-similar measure conjecture. Then…

Let $\mu$ be a spectral singular measure, and let a spectrum be a discrete set of frequencies whose exponential functions form an orthonormal basis for $L^2(\mu)$. The Beurling dim…

Randomised twisted Toeplitz spectral-measure conjecture. The measures $\mu_n$ converge weakly to

Banded perturbation conjecture. The measures $\mu_n$ converge weakly to

Let $(\mu,\Lambda)$ be a singular continuous spectral pair in $\mathbb{R}$. A real number $t\in\mathbb{R}$ is called a spectral eigenvalue of $(\mu,\Lambda)$ if $(\mu,t\Lambda)$ is…

Affirmative Beurling density conjecture. For every $(t,s)\in(0,\frac{1}{2}]\times[0,\infty)$, there exists a spectrum $Lambda:= Lambda{t,s}$ of $mu{4,{0,2}}$ such that

Sparse elliptic limiting-measure conjecture. For every $c>0$, there exists a deterministic measure $\mu_c$ that is the weak limit of the empirical spectral measures of the model wi…

Closed-curve non-spectrality conjecture. The measure $mu$ is not spectral.

Support conjecture for tight-frame spectra. Then

Infinite complete numbers conjecture. Every singular spectral pair $(\mu,\Lambda)$ has infinitely many complete numbers.

Let $d\ge 3$ be fixed, and let $\mathsf{A}_{\mathcal{G}}$ be the adjacency matrix of a uniformly random $d$-regular simple digraph on $N$ vertices, so that its entries are binary,…

Modified Łaba–Wang conjecture. If $\mu_{\rho,\mathcal{D},{\bf p}}$ is a spectral measure, then ${\bf p}$ is equal-weight, with $p_j=1/m$; $\rho=1/N$ for some integer $N\geq2$; and…

Let $N\ge 2$ be a positive integer, let $\mathcal{D}$ be a digit set defining the self-similar measure $mu_{N,\mathcal{D}}$, and let a product-form Hadamard triple with respect to…

Beurling-density upper-bound conjecture. For any maximal orthogonal set $\Lambda$ of $\mu_{p,q}$,

Exponential localization and symmetry conjecture. The densities of states $\nu^{\gamma,\beta}_{\mathrm{INB},+,k}$ and $\nu^{\gamma,\beta}_{\mathrm{INB},\times,k}$ exist and satisfy…

Let $f_\rho$ be a Gaussian stationary process with spectral measure $rho\in\mathcal{L}\cap\mathcal{M}$, and let $a<b$ with $a\ne-b$. A two-sided barrier event is the event that…

Let $ell\in\mathbb{R}$ and let $\rho,nu\in\mathcal{L}$ be spectral measures. Write $psi_\rho^\ell$ for the ball exponent and $theta_\rho^\ell$ for the persistence exponent. Theorem…