77 problems

Let $d\in\mathbb{N}$, let $\widehat{f}$ denote the unitary Fourier transform of $f\in\mathcal{L}^2(\mathbb{R}^d)$, and for $f\in\mathcal{L}^2(\mathbb{R}^d)\setminus\{0\}$ define it…

Cazacu–Flynn–Lam's conjecture. The sharp constant satisfies

Let $d\geq 1$, let $g\in L^2(\mathbb{R}^d)\setminus\{0\}$, and let $\Lambda=\{z_1,\ldots,z_N\}\subset\mathbb{R}^{2d}$ be a finite set of distinct points. For…

Let ${\cal H}_-^p$ be the odd subspace of $L^p(\mathbb{R})$, and let $(\psi_a)$ be a minimizing sequence for ${\cal S}_q$ as $a\to 0$ through positive values. Let $d_0$ be the dist…

Conjecture 2. The operator norm is

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 $0<\eta\leq\frac{1}{3}$ and let … for $(a_0,a_1,\ldots,a_N)\in\mathbb{C}^{N+1}$. The preceding theorem states that, modulo a trivial transformation, every solution of the ambig…

Sharp $L^p$ estimate conjecture. For any real $p\geq2$ and for any $h\in H$ there exists a constant $C(p)\leq1$, with $C(2)=1$, such that:

Let $PW=\{f\in L^2(\mathbb{R}):\operatorname{supp}\widehat{f}\subseteq[-\pi,\pi]\}$, and fix $s>0$. Consider all $f\in PW$ with $\|f\|_2=1$ and all measurable sets…

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

Strong finite-field Descartes rule. There exist constants $\eps,\delta<1$ such that, for all sufficiently large such primes, if $k\le q^\delta$, then $A(z)$ has at most $\eps q$ ro…

Let $d$ be the dimension, let $m$ be a positive integer, and let $g\in\mathbb{C}^d$ be a window satisfying the hypotheses of the strong two-window phase retrieval theorem. The curr…

Let $M$ be the compact manifold under consideration, let $f,g$ be functions on $M$, and let $I$ be a missing spectral window with complement $I^c$ consisting of the observed freque…

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…

Steinerberger's conjecture. The characteristic function of the interval is a global extremizer of this uncertainty principle. Steinerberger proved that the characteristic function…

Let $C(N)$ be the sharp constant in the second order Hydrogen uncertainty principle … For $N\in\{2,3\}$, the paper establishes $C(N)<(N+1)^2/4$. The dimension-four strict-bound con…

Let $V$ be a bounded potential on $\mathbb{R}^d$, and let $u$ be a nontrivial solution of … Suppose that $C$ and $\epsilon$ are positive constants. Kondrat'ev–Landis conjecture. Su…

Optimal-bound conjecture. The optimal state-dependent constant is

Let $F_n$ be the Fourier matrix of order $n$, and let $\mathbb Z_n^*$ denote the multiplicative group modulo $n$. A matrix is principally non-singular if every principal submatrix…

Let $A$ and $B$ be observables of a quantum system in state $\rho$. Write $V_\rho(A)$ and $V_\rho(B)$ for their variances, $[A,B]$ for their commutator, and $\|[A,B]\|_\rho$ for th…

Sharp Amrein–Berthier constant conjecture. The sharp constant in this inequality is of the form

Higher-dimensional annular improvement conjecture. This bound should be improvable to

Let $S$ and $\Sigma$ be measurable subsets of $\mathbb R^d$ with finite measure, and let $C(S,\Sigma)$ denote the strong annihilating-pair constant. Nazarov's conjecture. For…