5 problems

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 $L_1,\dots,L_m$ be a system of linear forms in $d$ variables over $\mathbb{F}_p^n$. Its true complexity is the least integer $k$ such that, for every $\eta>0$, there exists…

Gowers–Wolf conjecture. The true complexity of $\mathcal{L}$ is equal to the smallest $k$ such that the functions $L_1^{k+1},\dots,L_m^{k+1}$ are linearly independent. This conject…