Table of Contents
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 $f:\mathbb{Z}/N\mathbb{Z}\to \u2102$ 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]):
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 ${C}^{3}(G/\Gamma )\subseteq {(G/\Gamma )}^{8}$ somewhat analogous to the parallelepipeds in an abelian group.
There is a plentiful supply of maps $p:\mathbb{Z}/N\mathbb{Z}\to G/\Gamma $ which send cubes to cubes; that is, $p\left(c\right)\in {C}^{3}(G/\Gamma )$ for any $c\in {C}^{3}(\mathbb{Z}/N\mathbb{Z})$ (with $p$ applied pointwise).
The cubes on $G/\Gamma $ satisfy a corner constraint: given $c\in {C}^{3}(G/\Gamma )$, if we know ${c}_{000},\cdots ,{c}_{110}$ then the last vertex ${c}_{111}$ is uniquely determined by the others.
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 $F:G/\Gamma \to \u2102$ 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 $\u03f5=\u03f5(F,\delta )$ so that if $f:\mathbb{Z}/N\mathbb{Z}\to \u2102$ 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 ${\parallel f\parallel}_{{U}^{3}}\ge \u03f5$. 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 ${R}_{j}^{\left(i\right)}:G/\Gamma \to \u2102$, and some bounded continuous ${H}_{\text{err}}:{(G/\Gamma )}^{7}\to \u2102$ 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 ${\u2225{R}_{j}^{\left(i\right)}\circ p\u2225}_{{U}^{3}}\le {\u2225{R}_{j}^{\left(i\right)}\u2225}_{\infty}$ 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 $F:X\to \u2102$ 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 $\u03f5=\u03f5\left(\delta \right)$ such that if $f:\mathbb{Z}/N\mathbb{Z}\to \u2102$, $\left|f\right|\le 1$ has ${\parallel f\parallel}_{{U}^{3}}\ge \delta $ then $|\langle f,\varphi \rangle |\ge \u03f5$ 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 $F:X\to \u2102$ 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 $(X,{C}^{\u2022}\left(X\right))$ 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 $c\left(\omega \right),\phantom{\rule{4pt}{0ex}}\omega \in {\{0,1\}}^{l}$ 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 ${\mathcal{D}}_{s}\left(A\right)$ for the cubespace defined by requiring that $c\in {C}^{\ell}\left({\mathcal{D}}_{s}\left(A\right)\right)$ 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 ${\mathcal{D}}^{s}\left(A\right)$ is a nilspace of degree $s$ (see [GMV16a, Example A.9]). A more advanced example is given by the following construction of the
a filtration of degree $s$ if $[{G}_{i},{G}_{j}]\subseteq {G}_{i+j}$ 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 ${G}_{\u2022}$ 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 ${HK}^{\ell}\left({G}_{\u2022}\right)$ for each $\ell $ to be the subgroup of ${G}^{{\{0,1\}}^{\ell}}$ generated by the elements of the form ${\left[g\right]}_{F}$, where $F\subseteq {\{0,1\}}^{\ell}$ 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 ${G}^{{\{0,1\}}^{\ell}}$ 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 $(G,{HK}^{\u2022}\left({G}_{\u2022}\right))$ 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 ${G}_{\u2022}$ be a filtration of degree $s$ on $G$. Define the cubespace $HK\left({G}_{\u2022}\right)/\Gamma \triangleq (X,{C}^{\u2022}(G/\Gamma ))$ by
One can prove that $HK\left({G}_{\u2022}\right)/\Gamma $ is a nilspace of degree $s$ (see [GMV16a, Proposition 2.6]).
[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.
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