What are nilspaces?

A cubespace is a structure consisting of a compact metric space $X$, together with closed sets $C^k\left(X\right)\subseteq X^{2^k}$ for each integer $k\ge 0$, satisfying certain axioms that we will recall later. We think of $C^k\left(X\right)$ as determining when a collection of $2^k$ points of $X$ form a “$k$-cube”. The structure $(X,C^k\left(X\right))$ is further called a nilspace if it also satisfies certain extra rigidity conditions. The notion of nilspaces has its origins in the work of Host and Kra [HK08], where these objects appeared under the name of “parallelepiped structures”. The study of these objects was furthered by Antolín Camarena and Szegedy [ACS12] (for an exposition see [Can16a, Can16b]), who in the same work formulated a strong structure theorem for nilspaces. The gist of the structure theorem is that nilspaces arise as inverse limits of nilmanifolds $X=G/\Gamma$ (here $G$ is a nilpotent Lie group and $\Gamma \subset G$ is a discrete cocompact subgroup); i.e. they come from compact homogeneous spaces of nilpotent Lie groups.

The theory of nilspaces is a useful tool in the area of higher order Fourier analysis (see [Tao12, Gow17]), and in particular, forms the basis of Szegedy’s approach [Sze12] to proving the inverse theorem for the Gowers norms (another approach being due to Green, Tao and Ziegler [GTZ12]).

Nilspaces can also be used in topological dynamics and ergodic theory. For example together with Freddie Manners and Péter Varjú we used them to generalize a result of Host, Kra and Maass [HKM10] characterizing the largest pronilfactor of a minimal group action ([GMV16a, GMV16b, GMV16c]).

A fundamental concept in additive combinatorics and in higher order Fourier analysis is that of Gowers norms (introduced by Gowers in [Gow01]). Given a function $f:\mathbb{Z}/N\mathbb{Z}\to ℂ$ for $N$ a prime (say), the Gowers norm of order 3 ${\parallel f\parallel }_{U^3}$ is defined in terms of an average over cube or parallelepiped configurations in $\mathbb{Z}/N\mathbb{Z}$, e.g.

where $C^3(\mathbb{Z}/N\mathbb{Z})\subseteq {(\mathbb{Z}/N\mathbb{Z})}^8$ consists of all tuples

for $x,h_1,h_2,h_3\in \mathbb{Z}/N\mathbb{Z}$.

Following the Introduction of [GMV16a] quite closely we will (very) informally discuss why nilspaces of degree 2 are obstructions for $U^3$ to be “small”. For notational simplicity we focus only on Gowers norms of order 3, but the argument works for all orders. Let us start by sketching the proof of why a function which correlates with a nilsequence (see below) has “large” Gowers norm (following [GT08, Proposition 12.6]):

By a nilsequence on $\mathbb{Z}/N\mathbb{Z}$ we mean a function of the form $\varphi =F\circ p$ where $p$ is as above and $F:G/\Gamma \to ℂ$ is Lipschitz. We thus prove: Suppose $\varphi =F\circ p$ is a nilsequence with ${\parallel F\parallel }_{\infty }\le 1$ and $F\neg \equiv 0$ and let $\delta >0$. There exist $ϵ=ϵ(F,\delta )$ so that if $f:\mathbb{Z}/N\mathbb{Z}\to ℂ$ is such that $\left|f\right|\le 1$ and $\left|\langle f,\varphi \rangle \right|\triangleq |\frac{1}{N}{\sum }_{x=0}^{N-1}f\left(x\right)\varphi \left(x\right)|\ge \delta$, then ${\parallel f\parallel }_{U^3}\ge ϵ$. The key point is that the lower bound on ${\parallel f\parallel }_{U^3}$ depends only on the choice of $F$ (and so implicitly of $G/\Gamma$) and on $\delta$; not on $N$ or $p$.

[Proof sketch] By property (3), there is a closed subset $Y\subseteq {(G/\Gamma )}^7$ and a function $\tau :Y\to (G/\Gamma )$ such that

Hence we get a continuous function $F\circ \tau$ on $Y$. By the Tietze extension theorem, we can extend this to a bounded continuous function $H$ on ${(G/\Gamma )}^7$. Any continuous function on a product space can be approximated (up to a small error in $L^{\infty }$) by a finite sum of products of functions on the factors: that is, we can decompose

for some bounded continuous functions $R_j^{\left(i\right)}:G/\Gamma \to ℂ$, and some bounded continuous $H_{\text{err}}:{(G/\Gamma )}^7\to ℂ$ such that $\parallel H_{\text{err}}{\parallel }_{\infty }=o_{F;k\to \infty }\left(1\right)$.

Now, for any $x,h_1,h_2,h_3$ in $\mathbb{Z}/N\mathbb{Z}$ we have that $(p\left(x\right),p(x+h_1),p(x+h_2),p(x+h_1+h_2),\cdots )$ is in $C^3(G/\Gamma )$, and so

Thus

where

Since $|\langle f,\varphi \rangle |$ is bounded away from zero, we deduce that

is bounded away from zero for some $i\in \{1,\cdots ,k\}$ (after choosing $k$ appropriately in terms on $F$ and $\delta$). But this expression is a “Gowers inner product” of eight functions, and by the Gowers–Cauchy–Schwarz inequality (essentially multiple applications of Cauchy–Schwarz, see [Tao12, Exercise 1.3.19]), this quantity is bounded above by

and noting that ${∥R_j^{\left(i\right)}\circ p∥}_{U^3}\le {∥R_j^{\left(i\right)}∥}_{\infty }$ which is bounded, we get a lower bound on ${\parallel f\parallel }_{U^3}$ as required.

The key point is that the only properties of nilmanifolds, nilsequences etc. that we have used are those described in (1),(2) and (3) above. So we have in fact shown:

If $X$ is any compact metric space equipped with some suitable notion of “cubes” as in (1), having cube-preserving maps $p:\mathbb{Z}/N\mathbb{Z}\to X$ as in (2), and satisfying a corner constraint as in (3), then functions of the form $F\circ p$, where $F:X\to ℂ$ is continuous, obstruct Gowers uniformity on $\mathbb{Z}/N\mathbb{Z}$ in the sense of the above claim.

Let us refer to such a space informally for now as a “nil-object” (the formal notion of a nilspace will be introduced later). Then the above observation can be summarized as follows: Any “nil-object” is an obstruction to Gowers uniformity.

The inverse theorem for the Gowers norms states that given $\delta >0$ there exist $ϵ=ϵ\left(\delta \right)$ such that if $f:\mathbb{Z}/N\mathbb{Z}\to ℂ$, $\left|f\right|\le 1$ has ${\parallel f\parallel }_{U^3}\ge \delta$ then $|\langle f,\varphi \rangle |\ge ϵ$ for some nilsequence $\varphi$ whose “complexity” is bounded in terms of $\delta$. This tends to suggest that any “nil-object” $X$ must be very closely related to an actual nilmanifold $G/\Gamma$.

The conclusion of work of Szegedy [Sze12] is that it is possible to go in the other direction. He argues that all functions $f$ with ${\parallel f\parallel }_{U^3}$ somewhat large correlate with something of the form $F\circ p$ where $p:\mathbb{Z}/N\mathbb{Z}\to X$ and $F:X\to ℂ$ is continuous, for some space $X$ equipped with a notion of cubes, and some cube-preserving $p$, obeying some fairly reasonable additional axioms. Assuming this, we conclude: The class of all “nil-objects” corresponds precisely to the obstructions to Gowers uniformity.

Hence, the inverse theorem for the Gowers norms is essentially equivalent to classifying nil-objects $X$, showing in effect that they are all – essentially – nilmanifolds. This structural result is the goal of [ACS12] (this is also proven by somewhat different methods in [GMV16a, GMV16b, GMV16c].).

At this stage you are probably curious what are these “nil-objects” from the previous section. We list the axioms without further comments and in the next section give some examples. A map $\varphi =({\varphi }_1,...,{\varphi }_k):{\{0,1\}}^l\to {\{0,1\}}^k$ is called a morphism of discrete cubes if each coordinate function ${\varphi }_j({\omega }_1,...,{\omega }_l)$ equals to either 0, 1, ${\omega }_i$ or $1-{\omega }_i$ for some $1\le i\le l$. Let $X$ be a metric space and for each integer $l\ge 0$ let $C^l\left(X\right)\subset X^{{\{0,1\}}^l}$ be a closed set. We say that $(X,C^{•}\left(X\right))$ is a cubespace if $C^0=X$ and $c\circ \varphi \in C^l\left(X\right)$ for any morphism of discrete cubes $\varphi :{\{0,1\}}^l\to {\{0,1\}}^k$ and any $c\in C^k\left(X\right)$. We refer to this property as cube invariance. We call the elements of $C^l\left(X\right)$ cubes of order $l$ (in short $l$-cubes). We call the points $c\left(\omega \right),\omega \in {\{0,1\}}^l$ the vertices of $c$.

Let $X$ be a cubespace and let $f:X\to {\{0,1\}}^l\setminus \overrightarrow{\left\{1\right\}}$ be a map. We call $f$ an $l$-corner if ${f|}_{\{{\omega }_i=0\}}$ is an $(l-1)$-cube for all $1\le i\le l$. We say that the cubspace $X$ has $s$-completion if any $s$-corner $f$ can be extended to a $s$-cube, that is to say, if there is a cube $c\in C^s\left(X\right)$ such that ${c|}_{{\{0,1\}}^s\setminus \left\{\overrightarrow{1}\right\}}=f$. We say that $X$ is fibrant if it has $s$-completion for all $s$. We say that a cubespace $X$ has $(s+1)$-uniqueness, if $c_1{{|}_{{\{0,1\}}^{s+1}\setminus \left\{\overrightarrow{1}\right\}}=c_2|}_{{\{0,1\}}^{s+1}\setminus \left\{\overrightarrow{1}\right\}}$ imples $c_1=c_2$ for any two $(s+1)$-cubes $c_1,c_2\in C^{s+1}\left(X\right)$.

A cubespace $X$ is a nilspace of degree $s$ if it is fibrant and has $(s+1)$-uniqueness. We say that a cubespace $X$ is a nilspace if it is a nilspace of degree $s$ for some $s$.

A simple example is given by the following construction. Let $A$ be a compact abelian group. We write ${𝒟}_s\left(A\right)$ for the cubespace defined by requiring that $c\in C^{\ell }\left({𝒟}_s\left(A\right)\right)$ if and only if

holds for any morphism of discrete cubes $\varphi :{\{0,1\}}^{s+1}\to {\{0,1\}}^{\ell }$, where we write $\left|\omega \right|={\sum }_{1\le i\le s+1}{\omega }_i$ for $\omega \in {\{0,1\}}^{s+1}$. One can prove that ${𝒟}^s\left(A\right)$ is a nilspace of degree $s$ (see [GMV16a, Example A.9]). A more advanced example is given by the following construction of the