402 problems

Given $1 \leq p \leq \infty$, define the Bochner–Riesz means … The Bochner–Riesz conjecture. As soon as $\alpha > 0$, there is a range of exponents $p$ for which … for all…

Let … be the graph of a smooth function $h$, and define the extension operator … Write $q'=\frac{q}{q-1}$. Stein's restriction conjecture. If the Gaussian curvature of $\mathcal S$…

Cohn–Elkies lifting conjecture. Every optimal solution $f$ can be lifted to such a function $h$ with the same value of $r$ and

Let $X$ be uniformly distributed on the Boolean hypercube $\\{-1,1\\}^n$, and let $Y$ be obtained by transmitting $X$ through a memoryless binary symmetric channel with crossover p…

Fourier decay conjecture.

Let $f$ be a square-integrable function, that is, $f\in L^2$ on the relevant domain, and consider its Fourier series. Luzin's conjecture. The Fourier series of every square-integra…

Let $\Omega\subset\mathbb{R}^2$ be a convex body, and define its covariogram by … A convex body is determined by its covariogram if this function determines it among all convex bod…

Iosevich–Sawyer conjecture. For $p>2$, this condition is necessary and sufficient for the boundedness of $\mathcal{M}$ on $L^p$.

Let $f$ have $d$ continuous derivatives on each interval between its jump discontinuities, and suppose its Fourier data are used in Eckhoff's algebraic reconstruction method. Let…

Let $f:\{-1,1\}^{n}\rightarrow\{-1,1\}$ be a Boolean function. Its Fourier entropy is the Shannon entropy of its power spectrum $\{\hat{f}(S)^2:S\subseteq\{1,\ldots,n\}\}$, and its…

Let $\mathbb T$ be the circle, let $\varphi$ be a continuous self-mapping of $\mathbb T$, and let $A(\mathbb T)$ denote the Wiener algebra. The Beurling–Helson condition is that ……

Totik's conjecture. For almost every $x\in\mathbb{T}$,

Let $1<p<2$. A function in $L^p(\mathbb R)$ is a cyclic vector with respect to translations if the closed linear span of its translates is all of $L^p(\mathbb R)$. N. Wiener charac…

Fourier entropy–influence conjecture. There exists a constant $c>0$ such that, for every $n$ and every Boolean function $f$,

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…

Generic sparse phase retrieval conjecture. The signal $x$ is uniquely determined, up to unavoidable ambiguities, as long as

Friedgut–Kalai entropy-influence conjecture. There is an absolute constant $K$ such that, for every $n$ and every Boolean function $f:\{-1,1\}^n\to\{-1,1\}$,

Let $P_0$ be the polynomial appearing in the degree-2 vortex solution and let $(u_3,v_3)$ denote the kernel pair at Fourier mode $m=3$. Kernel rationality conjecture. The denominat…

Arnold's conjecture. The asymptotic behavior of this oscillatory integral is completely determined by the Newton polyhedron of $\phi$ in a so-called adapted coordinate system.

Let $\mathcal{H}_{\operatorname{e}}$ be the function space defined in the paper, and let … A pair $(\Lambda_0,\Lambda_0)$ is a Fourier uniqueness pair for…

Polylogarithmic growth conjecture. There exists $k>0$ such that

Let $\mu_\gamma$ be Gaussian multiplicative chaos on the unit circle with parameter $\gamma$, and define the Fourier dimension $\operatorname{dim}_F(\mu)$ as the largest $s$ such t…

Let $U_{1},U_{2},U_{3}\subset U$ be a $\nu$-disjoint triple of squares, meaning that each has side length comparable to $\nu$ and distinct squares are separated by at least $\nu$.…

Normalized local restriction conjecture. If $p\geq\frac{2n}{n-1}$, then for every $\varepsilon>0$ and $R>1$,

Let $s\in[0,1]$, and let $\sigma$ be an $(s,1)$-Frostman measure on the curve $\Gamma$. The sharp exponent in the $L^6$ Fourier-decay estimate … was conjectured to be the sharp Fou…