Table of Contents
A cubespace is a structure consisting of a compact metric space , together with closed sets for each integer , satisfying certain axioms that we will recall later. We think of as determining when a collection of points of form a “-cube”. The structure 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 (here is a nilpotent Lie group and 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 for a prime (say), the Gowers norm of order 3 is defined in terms of an average over cube or parallelepiped configurations in , e.g.
where consists of all tuples
Following the Introduction of [GMV16a] quite closely we will (very) informally discuss why nilspaces of degree 2 are obstructions for 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 (with suitable additional structure) there is also a notion of cubes on , given by a construction due to Host and Kra ([HK05, HK08]) - see details below. Specifically, suppose is a 2-step nilpotent Lie group and a discrete co-compact subgroup; then there is a closed subset somewhat analogous to the parallelepipeds in an abelian group.
There is a plentiful supply of maps which send cubes to cubes; that is, for any (with applied pointwise).
The cubes on satisfy a corner constraint: given , if we know then the last vertex is uniquely determined by the others.
By a nilsequence on we mean a function of the form where is as above and is Lipschitz. We thus prove: Suppose is a nilsequence with and and let . There exist so that if is such that and , then . The key point is that the lower bound on depends only on the choice of (and so implicitly of ) and on ; not on or .
[Proof sketch] By property (3), there is a closed subset and a function such that
Hence we get a continuous function on . By the Tietze extension theorem, we can extend this to a bounded continuous function on . Any continuous function on a product space can be approximated (up to a small error in ) by a finite sum of products of functions on the factors: that is, we can decompose
for some bounded continuous functions , and some bounded continuous such that .
Now, for any in we have that is in , and so
Since is bounded away from zero, we deduce that
is bounded away from zero for some (after choosing appropriately in terms on and ). 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 which is bounded, we get a lower bound on 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 is any compact metric space equipped with some suitable notion of “cubes” as in (1), having cube-preserving maps as in (2), and satisfying a corner constraint as in (3), then functions of the form , where is continuous, obstruct Gowers uniformity on 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 there exist such that if , has then for some nilsequence whose “complexity” is bounded in terms of . This tends to suggest that any “nil-object” must be very closely related to an actual nilmanifold .
The conclusion of work of Szegedy [Sze12] is that it is possible to go in the other direction. He argues that all functions with somewhat large correlate with something of the form where and is continuous, for some space equipped with a notion of cubes, and some cube-preserving , 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 , 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 is called a morphism of discrete cubes if each coordinate function equals to either 0, 1, or for some . Let be a metric space and for each integer let be a closed set. We say that is a cubespace if and for any morphism of discrete cubes and any . We refer to this property as cube invariance. We call the elements of cubes of order (in short -cubes). We call the points the vertices of .
Let be a cubespace and let be a map. We call an -corner if is an -cube for all . We say that the cubspace has -completion if any -corner can be extended to a -cube, that is to say, if there is a cube such that . We say that is fibrant if it has -completion for all . We say that a cubespace has -uniqueness, if imples for any two -cubes .
A cubespace is a nilspace of degree if it is fibrant and has -uniqueness. We say that a cubespace is a nilspace if it is a nilspace of degree for some .
A simple example is given by the following construction. Let be a compact abelian group. We write for the cubespace defined by requiring that if and only if
holds for any morphism of discrete cubes , where we write for . One can prove that is a nilspace of degree (see [GMV16a, Example A.9]). A more advanced example is given by the following construction of the
a filtration of degree if for all , adopting the convention that for all (e.g. the lower central series of a nilpotent group ). We write as a shorthand to denote a group equipped with a filtration and call it a filtered group.
We define the for each to be the subgroup of generated by the elements of the form , where is a face of codimension for some , and is the element of given by if and otherwise. It is not hard to verify that is a cubespace.
Let be a nilmanifold (that is is a nilpotent Lie group and is a discrete cocompact subgroup). Let be a filtration of degree on . Define the cubespace by
One can prove that is a nilspace of degree (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.
[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 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 -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.
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[Tao12] Terence Tao. Higher order Fourier analysis, volume 142. American Mathematical Soc., 2012.