27 problems

Let $n$ be a positive integer, let $g\in L^{2}(\mathbb{R}^{n})$ be nonzero, and for $x,\omega\in\mathbb{R}^{n}$ let $T_x$ and $M_\omega$ denote the translation and modulation opera…

Let $G$ be a finite Abelian group, let $V_g f$ denote the time-frequency representation appearing in the paper, let $\widehat f$ denote the Fourier transform of $f$, and write…

Let $x$) and $p$ be the canonical position and momentum operators, and let $E_0$ be the ground-state energy of the self-adjoint operator $|x|+|p|$, namely … The optimal constant…

Support cardinality conjecture. For $d\geq 1$, we have

Let $P$ be a locally finite poset, and let $\mu_P(x,y)$ denote its Möbius function. Say that $P$ has the Möbius uncertainty property when this property holds as defined in the pape…

Let $N,d$ be positive integers, let $f:\mathbb{Z}_N^d\to\mathbb{C}$ be a signal with support $E$ and Fourier support $\Sigma$, and let $\|\cdot\|_{U^k}$ denote the Gowers $U^k$ nor…

Let $n$ be a square-free positive integer, let $\zeta_n$ be an $n$-th root of unity, and consider the matrix $F_n=(\zeta_n^{ij})_{0\leq i,j\leq n-1}$. A principal submatrix is a ma…

Noncommutative Tao uncertainty principle. For every $x\in\mathcal{A}^p\setminus\{0\}$,

Formal uncertainty–duality conjecture. If $A$ and $B$ satisfy

Let $(\Omega,\mu)$ and $(\Delta,\nu)$ be measure spaces, let $\mathcal{H}$ be a Hilbert space, and let $\{\tau_\alpha\}_{\alpha\in\Omega}$ and $\{\omega_\beta\}_{\beta\in\Delta}$ b…

Coefficient-growth conjecture. The conclusion of the uncertainty theorem referred to as thref{UncertThmRSD} can be reached if merely

Measure minimization conjecture. Then $f$ is the unique solution of

Sharp Gaussian decay conjecture. If both quantities are finite, then

Polar duality conjecture. The ellipsoids $\Omega_X$ and $\Omega_P$ form a dual pair:

Wigderson–Wigderson conjecture. For all $q\neq 2$, the image of

Incomparability conjecture. The image of $F_q$ is all of $\mathbb{R}_{>0}$.

Structured discrete Hankel-transform conjecture. The following pairs are feasible:

Discrete Hankel-transform feasibility conjecture. If $(k,q)$ is $(s,\nu)$-feasible, then $(k+1,q)$ and $(k,q-1)$ are $(s,\nu)$-feasible. The function…

Discrete square-size conjecture.

Discrete Fourier-transform feasibility conjecture. If $(k,q)$ is $s$-feasible, then $(k+1,q)$ and $(k,q-1)$ are $s$-feasible. The function $q\mapsto\mathbb{A}_s^{\textup{disc}}(q)$…

Positive sign uncertainty conjecture.

Linear-programming and sign uncertainty conjecture.

Finite-plane uncertainty conjecture. One has

Let $\mathcal{A}_+(28)$ denote the class of radial Schwartz functions on $\mathbb{R}^{28}$ satisfying the positive Fourier-eigenfunction uncertainty conditions. Dimension-28 root-l…

For $s\in\{+1,-1\}$ and dimension $d$, let $\mathrm{A}_s(d)$ denote the corresponding optimal radius in the Fourier-eigenfunction uncertainty principle. Asymptotic equality conject…