19 problems

Let $f:\mathbb{F}_2^n\to\mathbb{C}$ be a $K$-approximate quadratic, meaning that $f$ has large $U^3$ norm in the sense used in the paper. A quadratic phase is a function of the for…

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…

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…

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…

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…

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,…

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…

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…

Inverse conjecture for the Gowers norms. The stated correlation with a bounded Lipschitz nilsequence should hold for every $s$, $\delta$, $N$, and $f$ satisfying the hypotheses.

Polynomial $U^3(\mathbb F_2^n)$-inverse conjecture. If

Inverse conjecture $\operatorname{GI}(s)$. Let $\delta>0$ and $s\geqslant 0$. Then there exists an $\varepsilon=\varepsilon_{\delta,s,\mathbb{F}}>0$ such that for every finite-dime…

Let $f:\mathbb{F}_2^n\to\mathbb{F}_2$ be a Boolean function, and let $\|f\|_{U^3}$ denote its Gowers $U^3$ norm. Polynomial inverse Gowers conjecture for the $U^3$ norm. If … then…

Let $f:\mathbb{Z}/N\mathbb{Z}\to\mathbb{C}$ be a $K$-approximate quadratic. An elementary $2$-step nilsequence is a sequence $F(g^nx_0)$, where $F:G/\Gamma\to\mathbb{C}$ is Lipschi…

Let $d\geq 1$ and let $\varepsilon>0$. For each $n\geq 1$, write $\mathbf{F}_2^n$ for the Boolean vector space, let $f:\mathbf{F}_2^n\to[-1,1]$, let $\|f\|_{U^d(\mathbf{F}_2^n)}$ d…