9 problems

Let $s\geqslant 1$. An $s$-step nilmanifold is a quotient $G/\Gamma=(G/\Gamma,d_{G/\Gamma})$ with smooth metric, and an $s$-step nilsequence is a sequence of the form…

Let $G/\Gamma$ be a $k$-step nilmanifold, let $T_g(x\Gamma)=xg\Gamma$, and let $(F(T_g^n\cdot x))_{n\in\mathbb{N}}$ be a $k$-step nilsequence. The Möbius orthogonality conjecture f…

Let $a\colon\mathbb N\to\mathbb Z$ be strictly increasing. The notions of being good for seminorm control and for degree-$(\ell+1)$ seminorm control along $\ell$-term arithmetic pr…

For $A\subseteq[N]$, define the dual function counting normalized 3-term arithmetic progressions with common difference $d$ by … A 2-step nilsequence is understood in the sense of…

A set $E\subset\mathbb{N}$ is called $k$-step-$I_0$ if it has the interpolation property for $k$-step nilsequences. Strict hierarchy conjecture. For every $k\in\mathbb{N}$, there e…

Let $s \geq 0$. Let $G \backslash \Gamma$ be an $s$-step nilmanifold. Let $F:G \backslash \Gamma \rightarrow \mathbb{C}$ be Lipschitz continuous and let…

Growth-assumption conjecture. In Theorems 4 and $4'$, the growth assumption on $(r_n)$ is superfluous.

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 $(s_n)$ be a sequence of integers, and let $\boldsymbol{\text{good for }\ell\text{-convergence}}$ mean that the sequence has the corresponding multiple ergodic convergence prop…