19 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 $s\geqslant 1$ 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…

Möbius and Nilsequences Conjecture. $\operatorname{MN}(s)$ holds for every $s\geqslant 1$; equivalently, the Main Theorem for $2$-step nilmanifolds has an analogue for $s$-step nil…

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…

Let a polynomial multiple correlation sequence be associated with a measure-preserving system, and suppose that in the single-transformation case it decomposes into a nilsequence a…

Frantzikinakis's conjecture. All multi-correlation sequences can be written as an integral combination of generalized nilsequences, using the definition of nilsequences that permit…

Let $(X,\mathcal{X},\mu,T)$ be a measure-preserving system, let $a_1,\ldots,a_m\in\mathbb{Z}$ and $f_1,\ldots,f_m\in L^\infty(X,\mu)$, and let $(\Psi(n))_{n\in\mathbb{Z}}$ be a nil…

Let $G$ be the filtered Lie group containing a polynomial sequence $g$, let $\Psi=(\psi_1,\ldots,\psi_t)$ be the system of linear forms under consideration, and let $g^\Psi$ denote…

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…

Let $s\geq 0$. Let $G/\Gamma$ be an $s$-step nilmanifold, where $G$ is a connected, simply connected nilpotent Lie group of step $s$ and $\Gamma$ is a cocompact discrete subgroup.…

Let $d_1,d_2$ be positive integers and let $f:\mathbb{Z}\times\mathbb{Z}\to\mathbb{C}$ be a sequence. Suppose that, uniformly in $n_2$, the map $n_1\mapsto f(n_1,n_2)$ is a Lipschi…

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

The inverse conjecture for the Gowers $U^{s+1}[N]$-norm. There exists a finite collection ${\mathcal M}_{s,\delta}$ of $s$-step nilmanifolds $G/\Gamma$, each equipped with a smooth…

Let $[N]=\{1,\ldots,N\}$, let $f:[N]\to\mathbb{C}$ be $1$-bounded, and let $\|f\|_{U^{s+1}}$ denote the Gowers norm. A degree $s$ polynomial nilsequence is a sequence of the form…

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…