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In the modern theory of additive combinatorics, a large role is played by the *Gowers uniformity norms* , where , is a finite abelian group, and is a function (one can also consider these norms in finite approximate groups such as instead of finite groups, but we will focus on the group case here for simplicity). These norms can be defined by the formula

where we use the averaging notation

for any non-empty finite set (with denoting the cardinality of ), and is the multiplicative discrete derivative operator

One reason why these norms play an important role is that they control various multilinear averages. We give two sample examples here:

We establish these claims a little later in this post.

In some more recent literature (e.g., this paper of Conlon, Fox, and Zhao), the role of Gowers norms have been replaced by (generalisations) of the *cut norm*, a concept originating from graph theory. In this blog post, it will be convenient to define these cut norms in the language of probability theory (using boldface to denote random variables).

Definition 2 (Cut norm)Let be independent random variables with ; to avoid minor technicalities we assume that these random variables are discrete and take values in a finite set. Given a random variable of these independent random variables, we define thecut normwhere the supremum ranges over all choices of random variables that are -bounded (thus surely), and such that does not depend on .

If , we abbreviate as .

Strictly speaking, the cut norm is only a cut semi-norm when , but we will abuse notation by referring to it as a norm nevertheless.

Example 3If is a bipartite graph, and , are independent random variables chosen uniformly from respectively, thenwhere the supremum ranges over all -bounded functions , . The right hand side is essentially the cut norm of the graph , as defined for instance by Frieze and Kannan.

The cut norm is basically an expectation when :

Example 4If , we see from definition thatIf , one easily checks that

where is the conditional expectation of to the -algebra generated by all the variables other than , i.e., the -algebra generated by . In particular, if are independent random variables drawn uniformly from respectively, then

Here are some basic properties of the cut norm:

Lemma 5 (Basic properties of cut norm)Let be independent discrete random variables, and a function of these variables.

- (i) (Permutation invariance) The cut norm is invariant with respect to permutations of the , or permutations of the .
- (ii) (Conditioning) One has
where on the right-hand side we view, for each realisation of , as a function of the random variables alone, thus the right-hand side may be expanded as

- (iii) (Monotonicity) If , we have
- (iv) (Multiplicative invariances) If is a -bounded function that does not depend on one of the , then
In particular, if we additionally assume , then

- (v) (Cauchy-Schwarz) If , one has
where is a copy of that is independent of and is the random variable

- (vi) (Averaging) If and , where is another random variable independent of , and is a random variable depending on both and , then

*Proof:* The claims (i), (ii) are clear from expanding out all the definitions. The claim (iii) also easily follows from the definitions (the left-hand side involves a supremum over a more general class of multipliers , while the right-hand side omits the multiplier), as does (iv) (the multiplier can be absorbed into one of the multipliers in the definition of the cut norm). The claim (vi) follows by expanding out the definitions, and observing that all of the terms in the supremum appearing in the left-hand side also appear as terms in the supremum on the right-hand side. It remains to prove (v). By definition, the left-hand side is the supremum over all quantities of the form

where the are -bounded functions of that do not depend on . We average out in the direction (that is, we condition out the variables ), and pull out the factor (which does not depend on ), to write this as

which by Cauchy-Schwarz is bounded by

which can be expanded using the copy as

Expanding

and noting that each is -bounded and independent of for , we obtain the claim.

Now we can relate the cut norm to Gowers uniformity norms:

Lemma 6Let be a finite abelian group, let be independent random variables uniformly drawn from for some , and let . ThenIf is additionally assumed to be -bounded, we have the converse inequalities

*Proof:* Applying Lemma 5(v) times, we can bound

where are independent copies of that are also independent of . The expression inside the norm can also be written as

so by Example 4 one can write (6) as

which after some change of variables simplifies to

which by Cauchy-Schwarz is bounded by

which one can rearrange as

giving (2). A similar argument bounds

by

which gives (3).

For (4), we can reverse the above steps and expand as

which we can write as

for some -bounded function . This can in turn be expanded as

for some -bounded functions that do not depend on . By Example 4, this can be written as

which by several applications of Theorem 5(iii) and then Theorem 5(iv) can be bounded by

giving (4). A similar argument gives (5).

Now we can prove Proposition 1. We begin with part (i). By permutation we may assume , then by translation we may assume . Replacing by and by , we can write the left-hand side of (1) as

where

is a -bounded function that does not depend on . Taking to be independent random variables drawn uniformly from , the left-hand side of (1) can then be written as

which by Example 4 is bounded in magnitude by

After many applications of Lemma 5(iii), (iv), this is bounded by

By Lemma 5(ii) we may drop the variable, and then the claim follows from Lemma 6.

For part (ii), we replace by and by to write the left-hand side as

the point here is that the first factor does not involve , the second factor does not involve , and the third factor has no quadratic terms in . Letting be independent variables drawn uniformly from , we can use Example 4 to bound this in magnitude by

which by Lemma 5(i),(iii),(iv) is bounded by

and then by Lemma 5(v) we may bound this by

which by Example 4 is

Now the expression inside the expectation is the product of four factors, each of which is or applied to an affine form where depends on and is one of , , , . With probability , the four different values of are distinct, and then by part (i) we have

When they are not distinct, we can instead bound this quantity by . Taking expectations in , we obtain the claim.

The analogue of the inverse theorem for cut norms is the following claim (which I learned from Ben Green):

Lemma 7 (-type inverse theorem)Let be independent random variables drawn from a finite abelian group , and let be -bounded. Then we havewhere is the group of homomorphisms is a homomorphism from to , and .

*Proof:* Suppose first that for some , then by definition

for some -bounded . By Fourier expansion, the left-hand side is also

where . From Plancherel’s theorem we have

hence by Hölder’s inequality one has for some , and hence

Conversely, suppose (7) holds. Then there is such that

which on substitution and Example 4 implies

The term splits into the product of a factor not depending on , and a factor not depending on . Applying Lemma 5(iii), (iv) we conclude that

The claim follows.

The higher order inverse theorems are much less trivial (and the optimal quantitative bounds are not currently known). However, there is a useful *degree lowering* argument, due to Peluse and Prendiville, that can allow one to lower the order of a uniformity norm in some cases. We give a simple version of this argument here:

Lemma 8 (Degree lowering argument, special case)Let be a finite abelian group, let be a non-empty finite set, and let be a function of the form for some -bounded functions indexed by . Suppose thatfor some and . Then one of the following claims hold (with implied constants allowed to depend on ):

- (i) (Degree lowering) one has .
- (ii) (Non-zero frequency) There exist and non-zero such that

There are more sophisticated versions of this argument in which the frequency is “minor arc” rather than “zero frequency”, and then the Gowers norms are localised to suitable large arithmetic progressions; this is implicit in the above-mentioned paper of Peluse and Prendiville.

*Proof:* One can write

and hence we conclude that

for a set of tuples of density . Applying Lemma 6 and Lemma 7, we see that for each such tuple, there exists such that

where is drawn uniformly from .

Let us adopt the convention that vanishes for not in , then from Lemma 5(ii) we have

where are independent random variables drawn uniformly from and also independent of . By repeated application of Lemma 5(iii) we then have

Expanding out and using Lemma 5(iv) repeatedly we conclude that

From definition of we then have

By Lemma 5(vi), we see that the left-hand side is less than

where is drawn uniformly from , independently of . By repeated application of Lemma 5(i), (v) repeatedly, we conclude that

where are independent copies of that are also independent of , . By Lemma 5(ii) and Example 4 we conclude that

with probability .

The left-hand side can be rewritten as

where is the additive version of , thus

Translating , we can simplify this a little to

If the frequency is ever non-vanishing in the event (9) then conclusion (ii) applies. We conclude that

with probability . In particular, by the pigeonhole principle, there exist such that

with probability . Expanding this out, we obtain a representation of the form

holding with probability , where the are functions that do not depend on the coordinate. From (8) we conclude that

for of the tuples . Thus by Lemma 5(ii)

By repeated application of Lemma 5(iii) we then have

and then by repeated application of Lemma 5(iv)

and then the conclusion (i) follows from Lemma 6.

As an application of degree lowering, we give an inverse theorem for the average in Proposition 1(ii), first established by Bourgain-Chang and later reproved by Peluse (by different methods from those given here):

Proposition 9Let be a cyclic group of prime order. Suppose that one has -bounded functions such thatfor some . Then either , or one has

We remark that a modification of the arguments below also give .

*Proof:* The left-hand side of (10) can be written as

where is the *dual function*

By Cauchy-Schwarz one thus has

and hence by Proposition 1, we either have (in which case we are done) or

Writing with , we conclude that either , or that

for some and non-zero . The left-hand side can be rewritten as

where and . We can rewrite this in turn as

which is bounded by

where are independent random variables drawn uniformly from . Applying Lemma 5(v), we conclude that

However, a routine Gauss sum calculation reveals that the left-hand side is for some absolute constant because is non-zero, so that . The only remaining case to consider is when

Repeating the above arguments we then conclude that

and then

The left-hand side can be computed to equal , and the claim follows.

This argument was given for the cyclic group setting, but the argument can also be applied to the integers (see Peluse-Prendiville) and can also be used to establish an analogue over the reals (that was first obtained by Bourgain).

Tamar Ziegler and I have just uploaded to the arXiv two related papers: “Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteoristic factors” and “polynomial patterns in primes“, with the former developing a “quantitative Bessel inequality” for local Gowers norms that is crucial in the latter.

We use the term “concatenation theorem” to denote results in which structural control of a function in two or more “directions” can be “concatenated” into structural control in a *joint* direction. A trivial example of such a concatenation theorem is the following: if a function is constant in the first variable (thus is constant for each ), and also constant in the second variable (thus is constant for each ), then it is constant in the joint variable . A slightly less trivial example: if a function is affine-linear in the first variable (thus, for each , there exist such that for all ) and affine-linear in the second variable (thus, for each , there exist such that for all ) then is a quadratic polynomial in ; in fact it must take the form

for some real numbers . (This can be seen for instance by using the affine linearity in to show that the coefficients are also affine linear.)

The same phenomenon extends to higher degree polynomials. Given a function from one additive group to another, we say that is of *degree less than * along a subgroup of if all the -fold iterated differences of along directions in vanish, that is to say

for all and , where is the difference operator

(We adopt the convention that the only of degree less than is the zero function.)

We then have the following simple proposition:

Proposition 1 (Concatenation of polynomiality)Let be of degree less than along one subgroup of , and of degree less than along another subgroup of , for some . Then is of degree less than along the subgroup of .

Note the previous example was basically the case when , , , , and .

*Proof:* The claim is trivial for or (in which is constant along or respectively), so suppose inductively and the claim has already been proven for smaller values of .

We take a derivative in a direction along to obtain

where is the shift of by . Then we take a further shift by a direction to obtain

leading to the *cocycle equation*

Since has degree less than along and degree less than along , has degree less than along and less than along , so is degree less than along by induction hypothesis. Similarly is also of degree less than along . Combining this with the cocycle equation we see that is of degree less than along for any , and hence is of degree less than along , as required.

While this proposition is simple, it already illustrates some basic principles regarding how one would go about proving a concatenation theorem:

- (i) One should perform induction on the degrees involved, and take advantage of the recursive nature of degree (in this case, the fact that a function is of less than degree along some subgroup of directions iff all of its first derivatives along are of degree less than ).
- (ii) Structure is preserved by operations such as addition, shifting, and taking derivatives. In particular, if a function is of degree less than along some subgroup , then any derivative of is also of degree less than along ,
*even if does not belong to*.

Here is another simple example of a concatenation theorem. Suppose an at most countable additive group acts by measure-preserving shifts on some probability space ; we call the pair (or more precisely ) a *-system*. We say that a function is a *generalised eigenfunction of degree less than * along some subgroup of and some if one has

almost everywhere for all , and some functions of degree less than along , with the convention that a function has degree less than if and only if it is equal to . Thus for instance, a function is an generalised eigenfunction of degree less than along if it is constant on almost every -ergodic component of , and is a generalised function of degree less than along if it is an eigenfunction of the shift action on almost every -ergodic component of . A basic example of a higher order eigenfunction is the function on the *skew shift* with action given by the generator for some irrational . One can check that for every integer , where is a generalised eigenfunction of degree less than along , so is of degree less than along .

We then have

Proposition 2 (Concatenation of higher order eigenfunctions)Let be a -system, and let be a generalised eigenfunction of degree less than along one subgroup of , and a generalised eigenfunction of degree less than along another subgroup of , for some . Then is a generalised eigenfunction of degree less than along the subgroup of .

The argument is almost identical to that of the previous proposition and is left as an exercise to the reader. The key point is the point (ii) identified earlier: the space of generalised eigenfunctions of degree less than along is preserved by multiplication and shifts, as well as the operation of “taking derivatives” even along directions that do not lie in . (To prove this latter claim, one should restrict to the region where is non-zero, and then divide by to locate .)

A typical example of this proposition in action is as follows: consider the -system given by the -torus with generating shifts

for some irrational , which can be checked to give a action

The function can then be checked to be a generalised eigenfunction of degree less than along , and also less than along , and less than along . One can view this example as the dynamical systems translation of the example (1) (see this previous post for some more discussion of this sort of correspondence).

The main results of our concatenation paper are analogues of these propositions concerning a more complicated notion of “polynomial-like” structure that are of importance in additive combinatorics and in ergodic theory. On the ergodic theory side, the notion of structure is captured by the *Host-Kra characteristic factors* of a -system along a subgroup . These factors can be defined in a number of ways. One is by duality, using the *Gowers-Host-Kra uniformity seminorms* (defined for instance here) . Namely, is the factor of defined up to equivalence by the requirement that

An equivalent definition is in terms of the *dual functions* of along , which can be defined recursively by setting and

where denotes the ergodic average along a Følner sequence in (in fact one can also define these concepts in non-amenable abelian settings as per this previous post). The factor can then be alternately defined as the factor generated by the dual functions for .

In the case when and is -ergodic, a deep theorem of Host and Kra shows that the factor is equivalent to the inverse limit of nilsystems of step less than . A similar statement holds with replaced by any finitely generated group by Griesmer, while the case of an infinite vector space over a finite field was treated in this paper of Bergelson, Ziegler, and myself. The situation is more subtle when is not -ergodic, or when is -ergodic but is a proper subgroup of acting non-ergodically, when one has to start considering measurable families of directional nilsystems; see for instance this paper of Austin for some of the subtleties involved (for instance, higher order group cohomology begins to become relevant!).

One of our main theorems is then

Proposition 3 (Concatenation of characteristic factors)Let be a -system, and let be measurable with respect to the factor and with respect to the factor for some and some subgroups of . Then is also measurable with respect to the factor .

We give two proofs of this proposition in the paper; an ergodic-theoretic proof using the Host-Kra theory of “cocycles of type (along a subgroup )”, which can be used to inductively describe the factors , and a combinatorial proof based on a combinatorial analogue of this proposition which is harder to state (but which roughly speaking asserts that a function which is nearly orthogonal to all bounded functions of small norm, and also to all bounded functions of small norm, is also nearly orthogonal to alll bounded functions of small norm). The combinatorial proof parallels the proof of Proposition 2. A key point is that dual functions obey a property analogous to being a generalised eigenfunction, namely that

where and is a “structured function of order ” along . (In the language of this previous paper of mine, this is an assertion that dual functions are uniformly almost periodic of order .) Again, the point (ii) above is crucial, and in particular it is key that any structure that has is inherited by the associated functions and . This sort of inheritance is quite easy to accomplish in the ergodic setting, as there is a ready-made language of factors to encapsulate the concept of structure, and the shift-invariance and -algebra properties of factors make it easy to show that just about any “natural” operation one performs on a function measurable with respect to a given factor, returns a function that is still measurable in that factor. In the finitary combinatorial setting, though, encoding the fact (ii) becomes a remarkably complicated notational nightmare, requiring a huge amount of “epsilon management” and “second-order epsilon management” (in which one manages not only scalar epsilons, but also function-valued epsilons that depend on other parameters). In order to avoid all this we were forced to utilise a nonstandard analysis framework for the combinatorial theorems, which made the arguments greatly resemble the ergodic arguments in many respects (though the two settings are still not equivalent, see this previous blog post for some comparisons between the two settings). Unfortunately the arguments are still rather complicated.

For combinatorial applications, dual formulations of the concatenation theorem are more useful. A direct dualisation of the theorem yields the following decomposition theorem: a bounded function which is small in norm can be split into a component that is small in norm, and a component that is small in norm. (One may wish to understand this type of result by first proving the following baby version: any function that has mean zero on every coset of , can be decomposed as the sum of a function that has mean zero on every coset, and a function that has mean zero on every coset. This is dual to the assertion that a function that is constant on every coset and constant on every coset, is constant on every coset.) Combining this with some standard “almost orthogonality” arguments (i.e. Cauchy-Schwarz) give the following Bessel-type inequality: if one has a lot of subgroups and a bounded function is small in norm for most , then it is also small in norm for most . (Here is a baby version one may wish to warm up on: if a function has small mean on for some large prime , then it has small mean on most of the cosets of most of the one-dimensional subgroups of .)

There is also a generalisation of the above Bessel inequality (as well as several of the other results mentioned above) in which the subgroups are replaced by more general *coset progressions* (of bounded rank), so that one has a Bessel inequailty controlling “local” Gowers uniformity norms such as by “global” Gowers uniformity norms such as . This turns out to be particularly useful when attempting to compute polynomial averages such as

for various functions . After repeated use of the van der Corput lemma, one can control such averages by expressions such as

(actually one ends up with more complicated expressions than this, but let’s use this example for sake of discussion). This can be viewed as an average of various Gowers uniformity norms of along arithmetic progressions of the form for various . Using the above Bessel inequality, this can be controlled in turn by an average of various Gowers uniformity norms along rank two generalised arithmetic progressions of the form for various . But for generic , this rank two progression is close in a certain technical sense to the “global” interval (this is ultimately due to the basic fact that two randomly chosen large integers are likely to be coprime, or at least have a small gcd). As a consequence, one can use the concatenation theorems from our first paper to control expressions such as (2) in terms of *global* Gowers uniformity norms. This is important in number theoretic applications, when one is interested in computing sums such as

or

where and are the Möbius and von Mangoldt functions respectively. This is because we are able to control global Gowers uniformity norms of such functions (thanks to results such as the proof of the inverse conjecture for the Gowers norms, the orthogonality of the Möbius function with nilsequences, and asymptotics for linear equations in primes), but much less control is currently available for local Gowers uniformity norms, even with the assistance of the generalised Riemann hypothesis (see this previous blog post for some further discussion).

By combining these tools and strategies with the “transference principle” approach from our previous paper (as improved using the recent “densification” technique of Conlon, Fox, and Zhao, discussed in this previous post), we are able in particular to establish the following result:

Theorem 4 (Polynomial patterns in the primes)Let be polynomials of degree at most , whose degree coefficients are all distinct, for some . Suppose that is admissible in the sense that for every prime , there are such that are all coprime to . Then there exist infinitely many pairs of natural numbers such that are prime.

Furthermore, we obtain an asymptotic for the number of such pairs in the range , (actually for minor technical reasons we reduce the range of to be very slightly less than ). In fact one could in principle obtain asymptotics for smaller values of , and relax the requirement that the degree coefficients be distinct with the requirement that no two of the differ by a constant, provided one had good enough local uniformity results for the Möbius or von Mangoldt functions. For instance, we can obtain an asymptotic for triplets of the form unconditionally for , and conditionally on GRH for all , using known results on primes in short intervals on average.

The case of this theorem was obtained in a previous paper of myself and Ben Green (using the aforementioned conjectures on the Gowers uniformity norm and the orthogonality of the Möbius function with nilsequences, both of which are now proven). For higher , an older result of Tamar and myself was able to tackle the case when (though our results there only give lower bounds on the number of pairs , and no asymptotics). Both of these results generalise my older theorem with Ben Green on the primes containing arbitrarily long arithmetic progressions. The theorem also extends to multidimensional polynomials, in which case there are some additional previous results; see the paper for more details. We also get a technical refinement of our previous result on narrow polynomial progressions in (dense subsets of) the primes by making the progressions just a little bit narrower in the case of the density of the set one is using is small.

This week I have been at a Banff workshop “Combinatorics meets Ergodic theory“, focused on the combinatorics surrounding Szemerédi’s theorem and the Gowers uniformity norms on one hand, and the ergodic theory surrounding Furstenberg’s multiple recurrence theorem and the Host-Kra structure theory on the other. This was quite a fruitful workshop, and directly inspired the various posts this week on this blog. Incidentally, BIRS being as efficient as it is, videos for this week’s talks are already online.

As mentioned in the previous two posts, Ben Green, Tamar Ziegler, and myself proved the following inverse theorem for the Gowers norms:

Theorem 1 (Inverse theorem for Gowers norms)Let and be integers, and let . Suppose that is a function supported on such thatThen there exists a filtered nilmanifold of degree and complexity , a polynomial sequence , and a Lipschitz function of Lipschitz constant such that

There is a higher dimensional generalisation, which first appeared explicitly (in a more general form) in this preprint of Szegedy (which used a slightly different argument than the one of Ben, Tammy, and myself; see also this previous preprint of Szegedy with related results):

Theorem 2 (Inverse theorem for multidimensional Gowers norms)Let and be integers, and let . Suppose that is a function supported on such thatThen there exists a filtered nilmanifold of degree and complexity , a polynomial sequence , and a Lipschitz function of Lipschitz constant such that

The case of this theorem was recently used by Wenbo Sun. One can replace the polynomial sequence with a linear sequence if desired by using a lifting trick (essentially due to Furstenberg, but which appears explicitly in Appendix C of my paper with Ben and Tammy).

In this post I would like to record a very neat and simple observation of Ben Green and Nikos Frantzikinakis, that uses the tool of Freiman isomorphisms to derive Theorem 2 as a corollary of the one-dimensional theorem. Namely, consider the linear map defined by

that is to say is the digit string base that has digits . This map is a linear map from to a subset of of density . Furthermore it has the following “Freiman isomorphism” property: if lie in with in the image set of for all , then there exist (unique) lifts such that

and

for all . Indeed, the injectivity of on uniquely determines the sum for each , and one can use base arithmetic to verify that the alternating sum of these sums on any -facet of the cube vanishes, which gives the claim. (In the language of additive combinatorics, the point is that is a Freiman isomorphism of order (say) on .)

Now let be the function defined by setting whenever , with vanishing outside of . If obeys (1), then from the above Freiman isomorphism property we have

Applying the one-dimensional inverse theorem (Theorem 1), with reduced by a factor of and replaced by , this implies the existence of a filtered nilmanifold of degree and complexity , a polynomial sequence , and a Lipschitz function of Lipschitz constant such that

which by the Freiman isomorphism property again implies that

But the map is clearly a polynomial map from to (the composition of two polynomial maps is polynomial, see e.g. Appendix B of my paper with Ben and Tammy), and the claim follows.

Remark 3This trick appears to be largely restricted to the case of boundedly generated groups such as ; I do not see any easy way to deduce an inverse theorem for, say, from the -inverse theorem by this method.

Remark 4By combining this argument with the one in the previous post, one can obtain a weak ergodic inverse theorem for -actions. Interestingly, the Freiman isomorphism argument appears to be difficult to implement directly in the ergodic category; in particular, there does not appear to be an obvious direct way to derive the Host-Kra inverse theorem for actions (a result first obtained in the PhD thesis of Griesmer) from the counterpart for actions.

Note: this post is of a particularly technical nature, in particular presuming familiarity with nilsequences, nilsystems, characteristic factors, etc., and is primarily intended for experts.

As mentioned in the previous post, Ben Green, Tamar Ziegler, and myself proved the following inverse theorem for the Gowers norms:

Theorem 1 (Inverse theorem for Gowers norms)Let and be integers, and let . Suppose that is a function supported on such thatThen there exists a filtered nilmanifold of degree and complexity , a polynomial sequence , and a Lipschitz function of Lipschitz constant such that

This result was conjectured earlier by Ben Green and myself; this conjecture was strongly motivated by an analogous inverse theorem in ergodic theory by Host and Kra, which we formulate here in a form designed to resemble Theorem 1 as closely as possible:

Theorem 2 (Inverse theorem for Gowers-Host-Kra seminorms)Let be an integer, and let be an ergodic, countably generated measure-preserving system. Suppose that one hasfor all non-zero (all spaces are real-valued in this post). Then is an inverse limit (in the category of measure-preserving systems, up to almost everywhere equivalence) of ergodic degree nilsystems, that is to say systems of the form for some degree filtered nilmanifold and a group element that acts ergodically on .

It is a natural question to ask if there is any logical relationship between the two theorems. In the finite field category, one can deduce the combinatorial inverse theorem from the ergodic inverse theorem by a variant of the Furstenberg correspondence principle, as worked out by Tamar Ziegler and myself, however in the current context of -actions, the connection is less clear.

One can split Theorem 2 into two components:

Theorem 3 (Weak inverse theorem for Gowers-Host-Kra seminorms)Let be an integer, and let be an ergodic, countably generated measure-preserving system. Suppose that one hasfor all non-zero , where . Then is a

factorof an inverse limit of ergodic degree nilsystems.

Theorem 4 (Pro-nilsystems closed under factors)Let be an integer. Then any factor of an inverse limit of ergodic degree nilsystems, is again an inverse limit of ergodic degree nilsystems.

Indeed, it is clear that Theorem 2 implies both Theorem 3 and Theorem 4, and conversely that the two latter theorems jointly imply the former. Theorem 4 is, in principle, purely a fact about nilsystems, and should have an independent proof, but this is not known; the only known proofs go through the full machinery needed to prove Theorem 2 (or the closely related theorem of Ziegler). (However, the fact that a factor of a nilsystem is again a nilsystem was established previously by Parry.)

The purpose of this post is to record a partial implication in reverse direction to the correspondence principle:

As mentioned at the start of the post, a fair amount of familiarity with the area is presumed here, and some routine steps will be presented with only a fairly brief explanation.

A few years ago, Ben Green, Tamar Ziegler, and myself proved the following (rather technical-looking) inverse theorem for the Gowers norms:

Theorem 1 (Discrete inverse theorem for Gowers norms)Let and be integers, and let . Suppose that is a function supported on such that

For the definitions of “filtered nilmanifold”, “degree”, “complexity”, and “polynomial sequence”, see the paper of Ben, Tammy, and myself. (I should caution the reader that this blog post will presume a fair amount of familiarity with this subfield of additive combinatorics.) This result has a number of applications, for instance to establishing asymptotics for linear equations in the primes, but this will not be the focus of discussion here.

The purpose of this post is to record the observation that this “discrete” inverse theorem, together with an equidistribution theorem for nilsequences that Ben and I worked out in a separate paper, implies a continuous version:

Theorem 2 (Continuous inverse theorem for Gowers norms)Let be an integer, and let . Suppose that is a measurable function supported on such thatThen there exists a filtered nilmanifold of degree and complexity , a (smooth) polynomial sequence , and a Lipschitz function of Lipschitz constant such that

The interval can be easily replaced with any other fixed interval by a change of variables. A key point here is that the bounds are completely uniform in the choice of . Note though that the coefficients of can be arbitrarily large (and this is necessary, as can be seen just by considering functions of the form for some arbitrarily large frequency ).

It is likely that one could prove Theorem 2 by carefully going through the proof of Theorem 1 and replacing all instances of with (and making appropriate modifications to the argument to accommodate this). However, the proof of Theorem 1 is quite lengthy. Here, we shall proceed by the usual limiting process of viewing the continuous interval as a limit of the discrete interval as . However there will be some problems taking the limit due to a failure of compactness, and specifically with regards to the coefficients of the polynomial sequence produced by Theorem 1, after normalising these coefficients by . Fortunately, a factorisation theorem from a paper of Ben Green and myself resolves this problem by splitting into a “smooth” part which does enjoy good compactness properties, as well as “totally equidistributed” and “periodic” parts which can be eliminated using the measurability (and thus, approximate smoothness), of .

Szemerédi’s theorem asserts that any subset of the integers of positive upper density contains arbitrarily large arithmetic progressions. Here is an equivalent quantitative form of this theorem:

Theorem 1 (Szemerédi’s theorem)Let be a positive integer, and let be a function with for some , where we use the averaging notation , , etc.. Then for we havefor some depending only on .

The equivalence is basically thanks to an averaging argument of Varnavides; see for instance Chapter 11 of my book with Van Vu or this previous blog post for a discussion. We have removed the cases as they are trivial and somewhat degenerate.

There are now many proofs of this theorem. Some time ago, I took an ergodic-theoretic proof of Furstenberg and converted it to a purely finitary proof of the theorem. The argument used some simplifying innovations that had been developed since the original work of Furstenberg (in particular, deployment of the Gowers uniformity norms, as well as a “dual” norm that I called the uniformly almost periodic norm, and an emphasis on van der Waerden’s theorem for handling the “compact extension” component of the argument). But the proof was still quite messy. However, as discussed in this previous blog post, messy finitary proofs can often be cleaned up using nonstandard analysis. Thus, there should be a nonstandard version of the Furstenberg ergodic theory argument that is relatively clean. I decided (after some encouragement from Ben Green and Isaac Goldbring) to write down most of the details of this argument in this blog post, though for sake of brevity I will skim rather quickly over arguments that were already discussed at length in other blog posts. In particular, I will presume familiarity with nonstandard analysis (in particular, the notion of a standard part of a bounded real number, and the Loeb measure construction), see for instance this previous blog post for a discussion.

The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if is a unitary operator on a Hilbert space , and is a vector in that Hilbert space, then one has

in the strong topology, where is the -invariant subspace of , and is the orthogonal projection to . (See e.g. these previous lecture notes for a proof.) The same proof extends to more general amenable groups: if is a countable amenable group acting on a Hilbert space by unitary transformations for , and is a vector in that Hilbert space, then one has

for any Folner sequence of , where is the -invariant subspace, and is the average of on . Thus one can interpret as a certain average of elements of the orbit of .

In a previous blog post, I noted a variant of this ergodic theorem (due to Alaoglu and Birkhoff) that holds even when the group is not amenable (or not discrete), using a more abstract notion of averaging:

Theorem 1 (Abstract ergodic theorem)Let be an arbitrary group acting unitarily on a Hilbert space , and let be a vector in . Then is the element in the closed convex hull of of minimal norm, and is also the unique element of in this closed convex hull.

I recently stumbled upon a different way to think about this theorem, in the additive case when is abelian, which has a closer resemblance to the classical mean ergodic theorem. Given an arbitrary additive group (not necessarily discrete, or countable), let denote the collection of finite non-empty multisets in – that is to say, unordered collections of elements of , not necessarily distinct, for some positive integer . Given two multisets , in , we can form the sum set . Note that the sum set can contain multiplicity even when do not; for instance, . Given a multiset in , and a function from to a vector space , we define the average as

Note that the multiplicity function of the set affects the average; for instance, we have , but .

We can define a directed set on as follows: given two multisets , we write if we have for some . Thus for instance we have . It is easy to verify that this operation is transitive and reflexive, and is directed because any two elements of have a common upper bound, namely . (This is where we need to be abelian.) The notion of convergence along a net, now allows us to define the notion of convergence along ; given a family of points in a topological space indexed by elements of , and a point in , we say that *converges* to along if, for every open neighbourhood of in , one has for sufficiently large , that is to say there exists such that for all . If the topological space is Hausdorff, then the limit is unique (if it exists), and we then write

When takes values in the reals, one can also define the limit superior or limit inferior along such nets in the obvious fashion.

We can then give an alternate formulation of the abstract ergodic theorem in the abelian case:

Theorem 2 (Abelian abstract ergodic theorem)Let be an arbitrary additive group acting unitarily on a Hilbert space , and let be a vector in . Then we havein the strong topology of .

*Proof:* Suppose that , so that for some , then

so by unitarity and the triangle inequality we have

thus is monotone non-increasing in . Since this quantity is bounded between and , we conclude that the limit exists. Thus, for any , we have for sufficiently large that

for all . In particular, for any , we have

We can write

and so from the parallelogram law and unitarity we have

for all , and hence by the triangle inequality (averaging over a finite multiset )

for any . This shows that is a Cauchy sequence in (in the strong topology), and hence (by the completeness of ) tends to a limit. Shifting by a group element , we have

and hence is invariant under shifts, and thus lies in . On the other hand, for any and , we have

and thus on taking strong limits

and so is orthogonal to . Combining these two facts we see that is equal to as claimed.

To relate this result to the classical ergodic theorem, we observe

Lemma 3Let be a countable additive group, with a F{\o}lner sequence , and let be a bounded sequence in a normed vector space indexed by . If exists, then exists, and the two limits are equal.

*Proof:* From the F{\o}lner property, we see that for any and any , the averages and differ by at most in norm if is sufficiently large depending on , (and the ). On the other hand, by the existence of the limit , the averages and differ by at most in norm if is sufficiently large depending on (regardless of how large is). The claim follows.

It turns out that this approach can also be used as an alternate way to construct the Gowers–Host-Kra seminorms in ergodic theory, which has the feature that it does not explicitly require any amenability on the group (or separability on the underlying measure space), though, as pointed out to me in comments, even uncountable abelian groups are amenable in the sense of possessing an invariant mean, even if they do not have a F{\o}lner sequence.

Given an arbitrary additive group , define a *-system* to be a probability space (not necessarily separable or standard Borel), together with a collection of invertible, measure-preserving maps, such that is the identity and (modulo null sets) for all . This then gives isomorphisms for by setting . From the above abstract ergodic theorem, we see that

in the strong topology of for any , where is the collection of measurable sets that are essentially -invariant in the sense that modulo null sets for all , and is the conditional expectation of with respect to .

In a similar spirit, we have

Theorem 4 (Convergence of Gowers-Host-Kra seminorms)Let be a -system for some additive group . Let be a natural number, and for every , let , which for simplicity we take to be real-valued. Then the expressionconverges, where we write , and we are using the product direct set on to define the convergence . In particular, for , the limit

converges.

We prove this theorem below the fold. It implies a number of other known descriptions of the Gowers-Host-Kra seminorms , for instance that

for , while from the ergodic theorem we have

This definition also manifestly demonstrates the cube symmetries of the Host-Kra measures on , defined via duality by requiring that

In a subsequent blog post I hope to present a more detailed study of the norm and its relationship with eigenfunctions and the Kronecker factor, without assuming any amenability on or any separability or topological structure on .

I’ve just finished writing the first draft of my third book coming out of the 2010 blog posts, namely “Higher order Fourier analysis“, which was based primarily on my graduate course in the topic, though it also contains material from some additional posts related to linear and higher order Fourier analysis on the blog. It is available online here. As usual, comments and corrections are welcome. There is also a stub page for the book, which at present does not contain much more than the above link.

Tanja Eisner and I have just uploaded to the arXiv our paper “Large values of the Gowers-Host-Kra seminorms“, submitted to Journal d’Analyse Mathematique. This paper is concerned with the properties of three closely related families of (semi)norms, indexed by a positive integer :

- The
*Gowers uniformity norms*of a (bounded, measurable, compactly supported) function taking values on a locally compact abelian group , equipped with a Haar measure ; - The Gowers uniformity norms of a function on a discrete interval ; and
- The Gowers-Host-Kra seminorms of a function on an ergodic measure-preserving system .

These norms have been discussed in depth in previous blog posts, so I will just quickly review the definition of the first norm here (the other two (semi)norms are defined similarly). The norm is defined recursively by setting

and

where . Equivalently, one has

Informally, the Gowers uniformity norm measures the extent to which (the phase of ) behaves like a polynomial of degree less than . Indeed, if and is compact with normalised Haar measure , it is not difficult to show that is at most , with equality if and only if takes the form almost everywhere, where is a polynomial of degree less than (which means that for all ).

Our first result is to show that this result is robust, uniformly over all choices of group :

Theorem 1 (-near extremisers)Let be a compact abelian group with normalised Haar measure , and let be such that and for some and . Then there exists a polynomial of degree at most such that , where is bounded by a quantity that goes to zero as for fixed .

The quantity can be described effectively (it is of polynomial size in ), but we did not seek to optimise it here. This result was already known in the case of vector spaces over a fixed finite field (where it is essentially equivalent to the assertion that the property of being a polynomial of degree at most is locally testable); the extension to general groups turns out to fairly routine. The basic idea is to use the recursive structure of the Gowers norms, which tells us in particular that if is close to one, then is close to one for most , which by induction implies that is close to for some polynomials of degree at most and for most . (Actually, it is not difficult to use cocycle equations such as (when ) to upgrade “for most ” to “for all “.) To finish the job, one would like to express the as derivatives of a polynomial of degree at most . This turns out to be equivalent to requiring that the obey the cocycle equation

where is the translate of by . (In the paper, the sign conventions are reversed, so that , in order to be compatible with ergodic theory notation, but this makes no substantial difference to the arguments or results.) However, one does not quite get this right away; instead, by using some separation properties of polynomials, one can show the weaker statement that

where the are small real constants. To eliminate these constants, one exploits the trivial cohomology of the real line. From (1) one soon concludes that the obey the -cocycle equation

and an averaging argument then shows that is a -coboundary in the sense that

for some small scalar depending on . Subtracting from then gives the claim.

Similar results and arguments also hold for the and norms, which we will not detail here.

Dimensional analysis reveals that the norm is not actually the most natural norm with which to compare the norms against. An application of Young’s convolution inequality in fact reveals that one has the inequality

where is the critical exponent , without any compactness or normalisation hypothesis on the group and the Haar measure . This allows us to extend the norm to all of . There is then a stronger inverse theorem available:

Theorem 2 (-near extremisers)Let be a locally compact abelian group, and let be such that and for some and . Then there exists a coset of a compact open subgroup of , and a polynomial of degree at most such that .

Conversely, it is not difficult to show that equality in (2) is attained when takes the form as above. The main idea of proof is to use an inverse theorem for Young’s inequality due to Fournier to reduce matters to the case that was already established. An analogous result is also obtained for the norm on an ergodic system; but for technical reasons, the methods do not seem to apply easily to the norm. (This norm is essentially equivalent to the norm up to constants, with comparable to , but when working with near-extremisers, norms that are only equivalent up to constants can have quite different near-extremal behaviour.)

In the case when is a Euclidean group , it is possible to use the sharp Young inequality of Beckner and of Brascamp-Lieb to improve (2) somewhat. For instance, when , one has

with equality attained if and only if is a gaussian modulated by a quadratic polynomial phase. This additional gain of allows one to pinpoint the threshold for the previous near-extremiser results in the case of norms. For instance, by using the Host-Kra machinery of characteristic factors for the norm, combined with an explicit and concrete analysis of the -step nilsystems generated by that machinery, we can show that

whenever is a *totally* ergodic system and is orthogonal to all linear and quadratic eigenfunctions (which would otherwise form immediate counterexamples to the above inequality), with the factor being best possible. We can also establish analogous results for the and norms (using the inverse theorem of Ben Green and myself, in place of the Host-Kra machinery), although it is not clear to us whether the threshold remains best possible in this case.

In Notes 5, we saw that the Gowers uniformity norms on vector spaces in high characteristic were controlled by classical polynomial phases .

Now we study the analogous situation on cyclic groups . Here, there is an unexpected surprise: the polynomial phases (classical or otherwise) are no longer sufficient to control the Gowers norms once exceeds . To resolve this problem, one must enlarge the space of polynomials to a larger class. It turns out that there are at least three closely related options for this class: the *local polynomials*, the *bracket polynomials*, and the *nilsequences*. Each of the three classes has its own strengths and weaknesses, but in my opinion the nilsequences seem to be the most natural class, due to the rich algebraic and dynamical structure coming from the nilpotent Lie group undergirding such sequences. For reasons of space we shall focus primarily on the nilsequence viewpoint here.

Traditionally, nilsequences have been defined in terms of linear orbits on nilmanifolds ; however, in recent years it has been realised that it is convenient for technical reasons (particularly for the quantitative “single-scale” theory) to generalise this setup to that of *polynomial* orbits , and this is the perspective we will take here.

A polynomial phase on a finite abelian group is formed by starting with a polynomial to the unit circle, and then composing it with the exponential function . To create a nilsequence , we generalise this construction by starting with a polynomial into a *nilmanifold* , and then composing this with a Lipschitz function . (The Lipschitz regularity class is convenient for minor technical reasons, but one could also use other regularity classes here if desired.) These classes of sequences certainly include the polynomial phases, but are somewhat more general; for instance, they *almost* include *bracket polynomial* phases such as . (The “almost” here is because the relevant functions involved are only piecewise Lipschitz rather than Lipschitz, but this is primarily a technical issue and one should view bracket polynomial phases as “morally” being nilsequences.)

In these notes we set out the basic theory for these nilsequences, including their equidistribution theory (which generalises the equidistribution theory of polynomial flows on tori from Notes 1) and show that they are indeed obstructions to the Gowers norm being small. This leads to the *inverse conjecture for the Gowers norms* that shows that the Gowers norms on cyclic groups are indeed controlled by these sequences.

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