# What are nilspaces?

## Introduction

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 $\left(X,{C}^{k}\left(X\right)\right)$ 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]).

## Obstructions to Gowers uniformity

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 $f:ℤ/Nℤ\to ℂ$ 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 $ℤ/Nℤ$, e.g.

${\parallel f\parallel }_{{U}^{3}}^{8}={𝔼}_{c\in {C}^{3}\left(ℤ/Nℤ\right)}f\left({c}_{000}\right)\overline{f\left({c}_{001}\right)f\left({c}_{010}\right)}f\left({c}_{011}\right)\overline{f\left({c}_{100}\right)}f\left({c}_{101}\right)f\left({c}_{110}\right)\overline{f\left({c}_{111}\right)}$

where ${C}^{3}\left(ℤ/Nℤ\right)\subseteq {\left(ℤ/Nℤ\right)}^{8}$ consists of all tuples

${\left({c}_{\omega }\right)}_{\omega \in {\left\{0,1\right\}}^{3}}=x+{\omega }_{1}{h}_{1}+{\omega }_{2}{h}_{2}+{\omega }_{3}{h}_{3}$

for $x,{h}_{1},{h}_{2},{h}_{3}\in ℤ/Nℤ$.

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]):

1. Given a nilmanifold $G/\Gamma$ (with suitable additional structure) there is also a notion of cubes on $G/\Gamma$, given by a construction due to Host and Kra ([HK05, HK08]) - see details below. Specifically, suppose $G$ is a 2-step nilpotent Lie group and $\Gamma$ a discrete co-compact subgroup; then there is a closed subset ${C}^{3}\left(G/\Gamma \right)\subseteq {\left(G/\Gamma \right)}^{8}$ somewhat analogous to the parallelepipeds in an abelian group.

2. There is a plentiful supply of maps $p:ℤ/Nℤ\to G/\Gamma$ which send cubes to cubes; that is, $p\left(c\right)\in {C}^{3}\left(G/\Gamma \right)$ for any $c\in {C}^{3}\left(ℤ/Nℤ\right)$ (with $p$ applied pointwise).

3. The cubes on $G/\Gamma$ satisfy a corner constraint: given $c\in {C}^{3}\left(G/\Gamma \right)$, if we know ${c}_{000},\cdots ,{c}_{110}$ then the last vertex ${c}_{111}$ is uniquely determined by the others.

By a nilsequence on $ℤ/Nℤ$ 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 $F¬\equiv 0$ and let $\delta >0$. There exist $ϵ=ϵ\left(F,\delta \right)$ so that if $f:ℤ/Nℤ\to ℂ$ is such that $|f|\le 1$ and $|〈f,\varphi 〉|\triangleq |\frac{1}{N}{\sum }_{x=0}^{N-1}f\left(x\right)\varphi \left(x\right)|\ge \delta$, 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 $Y\subseteq {\left(G/\Gamma \right)}^{7}$ and a function $\tau :Y\to \left(G/\Gamma \right)$ such that

${C}^{3}\left(G/\Gamma \right)=\left\{\left(\tau \left(y\right),y\right):y\in Y\right\}\phantom{\rule{4pt}{0ex}}.$

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 ${\left(G/\Gamma \right)}^{7}$. 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

$H\left({x}_{1},\cdots ,{x}_{7}\right)=\sum _{i=1}^{k}{R}_{1}^{\left(i\right)}\left({x}_{1}\right)\cdots {R}_{7}^{\left(i\right)}\left({x}_{7}\right)+{H}_{\text{err}}$

for some bounded continuous functions ${R}_{j}^{\left(i\right)}:G/\Gamma \to ℂ$, and some bounded continuous ${H}_{\text{err}}:{\left(G/\Gamma \right)}^{7}\to ℂ$ 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 $ℤ/Nℤ$ we have that $\left(p\left(x\right),p\left(x+{h}_{1}\right),p\left(x+{h}_{2}\right),p\left(x+{h}_{1}+{h}_{2}\right),\cdots \right)$ is in ${C}^{3}\left(G/\Gamma \right)$, and so

$\begin{array}{cc}\hfill \varphi \left(x\right)& =F\left(p\left(x\right)\right)\hfill \\ & =F\left(\tau \left(p\left(x+{h}_{1}\right),p\left(x+{h}_{2}\right),p\left(x+{h}_{1}+{h}_{2}\right),\cdots ,p\left(x+{h}_{1}+{h}_{2}+{h}_{3}\right)\right)\right)\hfill \\ & =\sum _{i=1}^{k}{R}_{1}^{\left(i\right)}\left(p\left(x+{h}_{1}\right)\right){R}_{2}^{\left(i\right)}\left(p\left(x+{h}_{2}\right)\right)\cdots {R}_{7}^{\left(i\right)}\left(p\left(x+{h}_{1}+{h}_{2}+{h}_{3}\right)\right)+{o}_{k\to \infty }\left(1\right)\phantom{\rule{4pt}{0ex}}.\hfill \end{array}$

Thus

$\varphi \left(x\right)={𝔼}_{{h}_{1},{h}_{2},{h}_{3}}\left({R}_{1}^{\left(i\right)}\left(p\left(x+{h}_{1}\right)\right){R}_{2}^{\left(i\right)}\left(p\left(x+{h}_{2}\right)\right)\cdots {R}_{7}^{\left(i\right)}\left(p\left(x+{h}_{1}+{h}_{2}+{h}_{3}\right)\right)\right)+{o}_{k\to \infty }\left(1\right)$

where

${𝔼}_{x,y,z}h\left(x,y,z\right)\triangleq \frac{1}{{N}^{3}}\sum _{x=0}^{N-1}\sum _{y=0}^{N-1}\sum _{z=0}^{N-1}h\left(x,y,z\right)$

Since $|〈f,\varphi 〉|$ is bounded away from zero, we deduce that

$\left|{𝔼}_{x,{h}_{1},{h}_{2},{h}_{3}}f\left(x\right)\overline{{R}_{1}^{\left(i\right)}\left(p\left(x+{h}_{1}\right)\right){R}_{2}^{\left(i\right)}\left(p\left(x+{h}_{2}\right)\right)\cdots {R}_{7}^{\left(i\right)}\left(p\left(x+{h}_{1}+{h}_{2}+{h}_{3}\right)\right)}\right|$

is bounded away from zero for some $i\in \left\{1,\cdots ,k\right\}$ (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

${\parallel f\parallel }_{{U}^{3}}{∥{R}_{1}^{\left(i\right)}\circ p∥}_{{U}^{3}}\cdots {∥{R}_{7}^{\left(i\right)}\circ p∥}_{{U}^{3}}$

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 $p:ℤ/Nℤ\to X$ 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 $ℤ/Nℤ$ 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 $f:ℤ/Nℤ\to ℂ$, $|f|\le 1$ has ${\parallel f\parallel }_{{U}^{3}}\ge \delta$ then $|〈f,\varphi 〉|\ge ϵ$ 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 $p:ℤ/Nℤ\to X$ 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].).

## The nilspace axioms

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 $\varphi =\left({\varphi }_{1},...,{\varphi }_{k}\right):{\left\{0,1\right\}}^{l}\to {\left\{0,1\right\}}^{k}$ is called a morphism of discrete cubes if each coordinate function ${\varphi }_{j}\left({\omega }_{1},...,{\omega }_{l}\right)$ 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 ${C}^{l}\left(X\right)\subset {X}^{{\left\{0,1\right\}}^{l}}$ be a closed set. We say that $\left(X,{C}^{•}\left(X\right)\right)$ is a cubespace if ${C}^{0}=X$ and $c\circ \varphi \in {C}^{l}\left(X\right)$ for any morphism of discrete cubes $\varphi :{\left\{0,1\right\}}^{l}\to {\left\{0,1\right\}}^{k}$ 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 $c\left(\omega \right),\phantom{\rule{4pt}{0ex}}\omega \in {\left\{0,1\right\}}^{l}$ the vertices of $c$.

Let $X$ be a cubespace and let $f:X\to {\left\{0,1\right\}}^{l}\setminus \stackrel{\to }{\left\{1\right\}}$ be a map. We call $f$ an $l$-corner if ${f|}_{\left\{{\omega }_{i}=0\right\}}$ is an $\left(l-1\right)$-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 ${c|}_{{\left\{0,1\right\}}^{s}\setminus \left\{\stackrel{\to }{1}\right\}}=f$. We say that $X$ is fibrant if it has $s$-completion for all $s$. We say that a cubespace $X$ has $\left(s+1\right)$-uniqueness, if ${c}_{1}{{|}_{{\left\{0,1\right\}}^{s+1}\setminus \left\{\stackrel{\to }{1}\right\}}={c}_{2}|}_{{\left\{0,1\right\}}^{s+1}\setminus \left\{\stackrel{\to }{1}\right\}}$ imples ${c}_{1}={c}_{2}$ for any two $\left(s+1\right)$-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 $\left(s+1\right)$-uniqueness. We say that a cubespace $X$ is a nilspace if it is a nilspace of degree $s$ for some $s$.

## Examples of nilspaces

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

$\sum _{\omega \in {\left\{0,1\right\}}^{s+1}}{\left(-1\right)}^{|\omega |}c\left(\varphi \left(\omega \right)\right)=0$

holds for any morphism of discrete cubes $\varphi :{\left\{0,1\right\}}^{s+1}\to {\left\{0,1\right\}}^{\ell }$, where we write $|\omega |={\sum }_{1\le i\le s+1}{\omega }_{i}$ for $\omega \in {\left\{0,1\right\}}^{s+1}$. 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

Host–Kra nilspaces. Let $G$ be a (metric) topological group. We call a chain of closed subgroups

$G={G}_{0}\supseteq {G}_{1}\supseteq {G}_{2}\supseteq ...\supseteq {G}_{s+1}=\left\{1\right\}$

a filtration of degree $s$ if $\left[{G}_{i},{G}_{j}\right]\subseteq {G}_{i+j}$ for all $i,j\ge 0$, adopting the convention that ${G}_{i}=\left\{1\right\}$ for all $i\ge s+1$ (e.g. the lower central series of a nilpotent group $G$). We write ${G}_{•}$ as a shorthand to denote a group $G$ equipped with a filtration $\left\{{G}_{i}\right\}$ and call it a filtered group.

We define the ${HK}^{\ell }\left({G}_{•}\right)$ for each $\ell$ to be the subgroup of ${G}^{{\left\{0,1\right\}}^{\ell }}$ generated by the elements of the form ${\left[g\right]}_{F}$, where $F\subseteq {\left\{0,1\right\}}^{\ell }$ is a face of codimension $i$ for some $1\le i\le \ell$, $g\in {G}_{i}$ and ${\left[g\right]}_{F}$ is the element of ${G}^{{\left\{0,1\right\}}^{\ell }}$ given by ${\left[g\right]}_{F}\left(\omega \right)=g$ if $\omega \in F$ and ${\left[g\right]}_{F}\left(\omega \right)=id$ otherwise. It is not hard to verify that $\left(G,{HK}^{•}\left({G}_{•}\right)\right)$ is a cubespace.

Let $X=G/\Gamma$ be a nilmanifold (that is $G$ is a nilpotent Lie group and $\Gamma \subset G$ is a discrete cocompact subgroup). Let ${G}_{•}$ be a filtration of degree $s$ on $G$. Define the cubespace $HK\left({G}_{•}\right)/\Gamma \triangleq \left(X,{C}^{•}\left(G/\Gamma \right)\right)$ by

${C}^{\ell }\left(G/\Gamma \right)\triangleq \left\{\omega ↦g\left(\omega \right).x|\phantom{\rule{0.166667em}{0ex}}g\in {HK}^{\ell }\left({G}_{•}\right),x\in X\right\}.$

One can prove that $HK\left({G}_{•}\right)/\Gamma$ is a nilspace of degree $s$ (see [GMV16a, Proposition 2.6]).

## References

[ACS12] Omar Antolín Camarena and Balázs Szegedy. Nilspaces, nilmanifolds and their morphisms. Preprint. http://arxiv.org/abs/1009.3825, 2012.

[Can16a] Pablo Candela. Notes on compact nilspaces. arXiv preprint arXiv:1605.08940, 2016.

[Can16b] Pablo Candela. Notes on nilspaces: algebraic aspects. arXiv preprint arXiv:1601.03693, 2016.

[GMV16a] Yonatan Gutman, Freddie Manners, and Péter P. Varjú. The structure theory of nilspaces I. Preprint. arxiv.org/abs/1605.08945, 2016.

[GMV16b] Yonatan Gutman, Freddie Manners, and Péter P. Varjú. The structure theory of nilspaces II: Representation as nilmanifolds. Preprint. arxiv.org/abs/1605.08948, 2016.

[GMV16c] Yonatan Gutman, Freddie Manners, and Péter P. Varjú. The structure theory of nilspaces III: Inverse limit representations and topological dynamics. Preprint. arxiv.org/abs/1605.08950, 2016.

[Gow01] William T Gowers. A new proof of Szemerédi’s theorem. Geometric and functional analysis, 11(3):465–588, 2001.

[Gow17] W Gowers. Generalizations of Fourier analysis, and how to apply them. Bulletin of the American Mathematical Society, 54(1):1–44, 2017.

[GT08] Ben Green and Terence Tao. An inverse theorem for the Gowers ${U}^{3}\left(G\right)$ norm. Proceedings of the Edinburgh Mathematical Society (Series 2), 51(01):73–153, 2008.

[GTZ12] Ben Green, Terence Tao, and Tamar Ziegler. An inverse theorem for the Gowers ${U}^{s+1}\left[N\right]$-norm. Ann. of Math. (2), 176(2):1231–1372, 2012.

[HK05] Bernard Host and Bryna Kra. Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2), 161(1):397–488, 2005.

[HK08] Bernard Host and Bryna Kra. Parallelepipeds, nilpotent groups and Gowers norms. Bull. Soc. Math. France, 136(3):405–437, 2008.

[HKM10] Bernard Host, Bryna Kra, and Alejandro Maass. Nilsequences and a structure theorem for topological dynamical systems. Adv. Math., 224(1):103–129, 2010.

[Sze12] Balázs Szegedy. On higher order Fourier analysis. Preprint. http://arxiv.org/abs/1203.2260, 2012.

[Tao12] Terence Tao. Higher order Fourier analysis, volume 142. American Mathematical Soc., 2012.