A few years ago, Ben Green, Tamar Ziegler, and myself proved the following (rather technical-looking) inverse theorem for the Gowers norms:
Then there exists a filtered nilmanifold of degree and complexity , a polynomial sequence , and a Lipschitz function of Lipschitz constant 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:
Then 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 .
We now prove Theorem 2. Firstly observe from Hölder’s inequality that the Gowers norm expression in the left-hand side of (1) is continuous in in the topology. As such, it suffices to prove the theorem for a dense class of , such as the Lipschitz-continuous , so long as the bounds remain uniform in . Thus, we may assume without loss of generality that is Lipschitz continuous.
Now let be a large integer (which will eventually be sent to infinity along a subsequence). As is Lipschitz continuous, the integral in (1) is certainly Riemann integrable, and so for sufficiently large (where we allow “sufficiently large” to depend on the Lipschitz constant) we will have
(say). Applying Theorem 2, we can thus find for sufficiently large , a filtered nilmanifold of degree and complexity , a polynomial sequence , and a Lipschitz function of Lipschitz constant such that
Now we prepare to take limits as , passing to subsequences as necessary. Using Mal’cev bases, one can easily check that there are only finitely many filtered nilmanifolds of a fixed degree and complexity, hence by passing to a subsequence of we may assume that is independent of . The Lipschitz functions are now equicontinuous on a fixed compact domain , so by the Arzelá-Ascoli theorem and further passage to a subsequence we may assume that converges uniformly to a Lipschitz function of Lipschitz constant . In particular (passing to a further subsequence as necessary) we have
We have removed a lot of the dependencies of the nilsequence on , however there is still a serious lack of compactness in the remaining dependency of the polynomial sequence on . Fortunately, we can argue as follows. Let be large quantities (depending on , and the Lipschitz constant of ) to be chosen later. Applying the factorisation theorem for polynomial sequences (see Theorem 1.19 of this paper of Ben Green and myself), we may find for each in the current subsequence, an integer , a rational subgroup of whose associated filtered nilmanifold has structure constants that are -rational with respect to the Mal’cev bais of , and a decomposition
- is a polynomial sequence which is -smooth;
- is a polynomial sequence with is totally -equidistributed in ; and
- is a polynomial sequence which is -rational, and is periodic with period at most .
See the above referenced paper for a definition of all the terminology used here.
Once again we can make a lot of the data here independent of by passing to a subsequence. Firstly, takes only finitely many values so by passing to a subsequence we may assume that is independent of . Then the number of rational subgroups with -rational structure constants is also finite, so by passing to a further subsequence we may take independent of , so is also independent of . Up to right multiplication by polynomial sequences from to (which do not affect the value of ), there are only finitely many -rational polynomial sequences that are periodic with period at most , so we may take independent of . Finally, using coordinates one can write where is a continuous polynomial sequence whose coefficients are bounded uniformly in . By Bolzano-Weierstrass, we may assume on passing to a subsequence that converges locally uniformly to a limit , which is again a continuous polynomial sequence. Thus, on passing to a further subsequence, we have
Because is totally -equidistributed in , and is -rational, the conjugate is totally -equidistributed in , where and ; see Section 2 of this paper of Ben and myself for a derivation of this fact. From this, we have the approximation
for any and any fixed interval , where is the length of and the integral is with respect to Haar measure, and are sufficiently large depending on . Using Riemann integration, we thus see that the left-hand side of (2) is thus of the form
for sufficiently large if are sufficiently large depending on , and the Lipschitz constant of , and so (writing ) we have
if are sufficiently large depending on , and the Lipschitz constant of . If we let be a continuous polynomial sequence that is equidistributed in (which will happen as soon as the sequence is equidistributed with respect to the abelianisation of , by an old result of Leon Green), then a similar argument shows that
and thus there exists such that
Setting , we obtain the claim.
I thank Ben Green for helpful conversations that inspired this post.