150 problems

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 ……

Logarithmic Fourier-norm conjecture. If, for every finite subgroup $V\leq\widehat G$ with $|V|\leq M$, one has

Let $d\geq 2$, let $U$ be a compact neighbourhood of the origin in $\mathbb{R}^{d-1}$, and let $\Sigma:U\to\mathbb{R}^d$ be a smooth parametrisation of a $(d-1)$-dimensional subman…

For $d\geq1$, let $C_n$ be the collection of odd, transitive-symmetric functions $f:\{-1,1\}^n\to\{-1,1\}$. Let $\widehat f(S)$ denote Fourier coefficients, and let…

Rational-function lower-bound conjecture. If

Patterson–Wiedemann conjecture.

Self-dual relaxed lattice conjecture. In every dimension, the largest possible value of $r_1$ among self-dual relaxed lattices equals the smallest value of $r$ possible in Theorem…

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…

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 $\mathcal S$…

Friedgut–Kahn–Kalai–Keller spectral correlation conjecture. For any increasing Boolean functions $f,g:\{0,1\}^n\to\{0,1\}$,

Affine Fourier-spectrum threshold conjecture. If, for some $\theta\in[0,1]$,

Let $\Lambda_N^{\mathrm{sph}}$ denote the spherical truncation and let $S^{\mathrm{sph}}(N)$ be its associated incidence quantity. Spherical truncation incidence-growth conjecture.…

Let $X\sim\mu$, where $\mu$ is a log-concave measure on $\mathbb{R}^n$, and let $f:\mathbb{R}^n\to\mathbb{R}$ be a polynomial of degree $d$. For a Borel measure $\nu$ on…

Let $A\subset\mathbb{Z}$ be a finite set of size $N$, and define $\dim A$ to be the size of the largest dissociated subset of $A$. Let $K>0$. The inverse Littlewood dimension conje…

Let $A\subset\mathbb{Z}$ be a finite set of size $N$, and let $K>0$. Write $1_A$ for its indicator and define … A finite arithmetic progression is a finite subset of $\mathbb{Z}$ w…

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 $f_n(x)$ be the Fourier interpolation basis functions. Logarithmic square-norm conjecture. For $n>1$, one has … The lower bound is already known from the result cited in the pa…

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

Let $f_n(x)$ be the Fourier interpolation basis functions, and let $\varepsilon,\delta>0$. Consider the region $\varepsilon n\le x\le n-\sqrt n$. Interior-region decay conjecture.…

Let $f_n(x)$ denote the basis functions of the Fourier interpolation considered in the paper, for $n\in\mathbb{Z}_{\ge0}$ and $x\ge0$. Uniform boundedness conjecture. There exists…

Weighted Paley–Wiener conjecture. The following equivalences hold:

Let $f:(\{0,1\}^n,\mu_p^n)\to\{\pm1\}$ be Boolean-valued and let $0<p<1$. Define … and let $h(q)=-q\log q-(1-q)\log(1-q)$ be the binary entropy function. Binary-entropy lower-bound…