7 problems

For every even integer $k\ge 4$, let $\rho_k(\alpha)$ denote the minimum asymptotic density of $k$-term arithmetic progressions in a $U^{k-2}$-uniform set of density $\alpha$ (with…

Let $L_1,\dots,L_m$ be a system of linear forms, and let $\operatorname{true\ complexity}(L_1,\dots,L_m)$ denote the least integer $k$ such that the $U^{k+1}$ norm controls the ass…

Uniform sets bounded-sums conjecture. If $A$ is $\mathsf{U}^k(\Phi)$-uniform for some Følner sequence $\Phi$, then for every $\ell_1,\ldots,\ell_k\in\mathbb{N}$ there is an infinit…

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…

Gowers' conjecture. The two averages are close whenever $\\|f-g\\|_{U^k}$ is small.

Let $p$ be a prime, let $k$ be a positive integer, and let $f:\mathbb{F}_p^n\rightarrow\mathbb{C}$ be a function bounded in magnitude by $1$. Suppose that $f$ has large $U^{k+1}$ n…

Let $s\geqslant 1$, let $N$ be a positive integer, and let $f:\mathbb{Z}/N\mathbb{Z}\to[-1,1]$ satisfy … An $s$-step nilmanifold is a quotient $G/\Gamma$ of an $s$-step nilpotent L…