41 problems

Let $\phi_1,\dots,\phi_m\in\mathbf{Z}[x_1,\dots,x_d]$ be a system of linear forms in $d$ variables, and let $s$ denote its true complexity: the least natural number for which small…

Gowers–Wolf conjecture. These averages may be controlled by the Gowers $U^{s+1}$-norm of $f$.

Polynomial Gowers inverse conjecture. For every fixed $d$, the function $\Phi$ can be chosen to satisfy

Let $p$ be a prime and let $k\geq 1$. For a finite abelian group $G$, define the Gowers uniformity norm by … Here $\mathcal{C}$ denotes complex conjugation,…

Let $\mathbb{F}$ be a finite field, let $V$ be a finite-dimensional vector space over $\mathbb{F}$, and let $\mathcal{D}:= \{z \in \mathbb{C}: |z| \leqslant 1\}$ be the compact uni…

Let $s\geqslant 1$ and $0<\delta\leqslant 1$. An $s$-step nilmanifold is a quotient $G/\Gamma=(G/\Gamma,d_{G/\Gamma})$ equipped with a smooth metric, and an $s$-step nilsequence is…

Let $N$ be a large prime and let $f: \mathbb{Z}/N\mathbb{Z} \rightarrow [-1,1]$ satisfy $\|f\|_{U^3} \geqslant \delta$. Here $\|f\|_{U^3}$ denotes the Gowers uniformity norm, and…

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…

Original inverse Gowers conjecture. For every such $F$, a lower bound on the $U^{d+1}$ norm implies correlation with a phase polynomial of degree $d$, with correlation bounded belo…

Shao's conjecture. The leftmost expression in Shao's bound is the correct asymptotics of $t_{2,n}$ as $n\to\infty$; that is,

Let $\phi_1,\dots,\phi_k$ be a translation-invariant system of $\mathbb{F}_p$-linear maps $\mathbb{F}_p^n\to\mathbb{F}_p^n$ of complexity at most two. Let $\psi_1,\dots,\psi_s$ be…

Let $P_1,\dots,P_m\in\mathbf{Z}[y]$ be polynomials and let $0\leq i\leq m$. Consider the polynomial progression … An algebraic relation among these entries is an identity … where…

Let $k\geq4$ and let $\mathbf{a}=(a_1,\ldots,a_k)$ have increasing integer coordinates. An $\mathbf{a}$-AP is $n+a_1d,\ldots,n+a_kd$, and $\rho_{\mathbf a}(\alpha)$ denotes the cor…

For each integer $k\geq4$ and $0<\alpha<1$, let $\rho_k(\alpha)$ be the largest real number such that sufficiently $U^{k-2}$-uniform subsets of $\mathbb{Z}/N\mathbb{Z}$ of density…

For a finite abelian group, consider the inverse theorem for the corresponding Gowers norm and the correlating harmonics it produces. Jamneshan–Tao conjecture. The underlying objec…

Let $p$ be prime, $k\geq1$, $G=\mathbb{F}_p^n$, and $\mathbb D=\{z\in\mathbb C:|z|\leq1\}$. Let $\varepsilon:\mathbb N\to\mathbb R^+$ be decreasing, and let $\mathrm{Poly}^k(G)$ be…

Correlation-to-symmetric-rank conjecture. Then

Let $G$ be a finite additive group, let $ta > 0$, let $s eq 1$, and let $f \colon G \to \mathbb{C}$ be a $1$-bounded function with $\|f\|_{U^{s+1}(G)} \geq \eta$. A degree $s$ filt…

Let $\Phi=(\phi_1,\dots,\phi_k)$ be a system of linear forms, with $\phi_i\colon\mathbb{F}_p^d\to\mathbb{F}_p$, and let $i_0\in[k]$. Let $s(\Phi,i_0)$ be the asymmetric true comple…

Let $p$ be a prime and let $k\geq p+2$ be an integer. For each $n$, consider functions $f\colon\mathbb F_p^n\to\mathbb C$ with $\|f\|_{\infty}\leq 1$ and Gowers norm…

Let $\mathbf k=\mathbb F_p$ with $d>p$, let $V$ be a $\mathbf k$-vector space, and let $\psi$ be the additive character appearing in the correlation expression. For a function…

Kirshner–Samorodnitsky conjecture. One has

Let $a(n)$ be a $2$-automatic sequence such that … for every $q\in\mathbb{N}$ and $r\in\mathbb{N}_0$. Then the automatic-sequence uniformity conjecture. … for every…

For a finitely supported function $g\colon\mathbb{Z}\to\mathbb{C}$ and $d\geq 1$, define its Gowers norm by … where $|\vec\omega|=\omega_1+\dots+\omega_d$ and…

Finite-field inverse conjecture. There exists a polynomial $P:\mathbb F_p^n\to\mathbb F_p$ of degree at most $k$ such that