156 problems

Determine the set of thresholds $\delta>0$ for which there exists a constant $C_M(\delta)<\infty$, independent of the locally compact abelian group $G$, such that every measure…

For $n\ge 2$, let $K\subset\mathbb{R}^n$ be a centrally symmetric convex body, meaning $K=-K$, and let $B\subset\mathbb{R}^n$ be a Euclidean ball satisfying…

Let $M$ be a compact connected orientable Riemannian surface with boundary, and suppose that its boundary is isometric, with respect to the induced distance in $M$, to the circle o…

For every integer $m\ge 3$, suppose that the positive integers admit a partition $\mathbb{Z}_{>0}=B(\alpha_1,\beta_1)\sqcup\cdots\sqcup B(\alpha_m,\beta_m)$ into Beatty sequences…

For every finite Latin quandle $Q$, let $\mathbb{Z}[Q]$ be the free abelian group with basis $Q$, equipped with the bilinear multiplication extending the quandle operation $x*y$ on…

For a fixed dimension $d\ge 2$, do there exist arbitrarily large integers $N$ and lattices $\Lambda_1,\ldots,\Lambda_N\subset\mathbb{R}^d$ such that…

Quantum Fourier entropy-influence conjecture. There is a constant $c$ independent of $n$ such that

Let $r_1,r_2,\ldots$ be the positive real numbers of the form … where $j,k$ are integers. A radial Schwartz function is a function $f:\mathbf{R}^2\to\mathbf{R}$ invariant under rot…

Let $\gamma\in(1/\sqrt{2},\sqrt{2})$, let $\mu_\gamma$ be the Gaussian multiplicative chaos measure on $[0,1]$ with parameter $\gamma$, and let $c_n$ be its $n$-th Fourier coeffici…

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 $\mathcal S$…

Let $B_{d,n}(r)$ and $\widetilde{B}_{d,n}(r)$ be the functions appearing in the radial Fourier interpolation formulas for dimensions $d=3,4$, with $n$ an appropriate positive integ…

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

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…

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 $\nu$.…

Fourier-decay conjecture. There exists $\sigma>0$ such that

Fourier-Min-Entropy-Influence conjecture. There exists a constant $c>0$ such that, for every Boolean function $f:\\{-1,1\\}^n\to\\{-1,1\\}$,

Fourier-Entropy-Influence conjecture. There exists a constant $c>0$ such that, for every Boolean function $f:\\{-1,1\\}^n\to\\{-1,1\\}$,

Friedgut–Kalai Fourier entropy-influence conjecture. There exists a universal constant $c$ such that, for every such $f$,

Bourgain–Demeter–Kemp decoupling conjecture. There is a partition $\mathcal P_\delta=\mathcal P_\delta(\phi)$ of $[-1,1]^2$ into $\delta$-flat subsets $S$ such that, for every $f$…

Rudin's conjecture.

Linear adjoint cone restriction conjecture. The inequality above holds with constants depending on $S$, $n$ and $p,q$ if and only if $q>\frac{2n}{n-1}$ and…

Fix a degree $d$ and consider the torus $\mathcal H_d$ of hedgehogs whose support functions are trigonometric polynomials of degree $d$, with the amplitudes of each harmonic fixed.…

The short interval conjecture. For every $\alpha<1$, every such trigonometric polynomial satisfies

Let $G=\mathbb{F}_2^n$, let $f\in A(G)$, and let $\epsilon\in(0,1]$. Write … Here $\operatorname{codim}V$ denotes the codimension of a subspace $V\leq G$, and…

Let $p$ be a prime number and let $A\subset\mathbb{Z}/p\mathbb{Z}$ have density bounded away from $0$ and $1$ by an absolute constant. Green–Konyagin–Littlewood conjecture. Then ……