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.
Then 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:
for 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:
for all non-zero , where . Then is a factor of 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.
To show that is a factor of another system up to almost everywhere equivalence, it suffices to obtain a unital algebra homomorphism from to that intertwines with , and which is measure-preserving (or more precisely, integral-preserving). On the other hand, by hypothesis, is generated (as a von Neumann algebra) by the dual functions
for , where
indeed we may restrict to a countable sequence that is dense in in the (say) topology, together with their shifts. To obtain such a factor representation, it thus suffices to find a “model” associated to each dual function in such a fashion that
We may normalise all the to take values in . For any , we can find a scale such that
If we then define the exceptional set
for “most” .
Next, we observe from the Cauchy-Schwarz-Gowers inequality that for almost every , the dual function is anti-uniform in the sense that
for any function . By the usual structure theorems (e.g. Theorem 1.2 of this paper of Ben Green and myself) this shows that for almost every and every , there exists a degree nilsequence of complexity such that
(say). (Sketch of proof: standard structure theorems give a decomposition of the form
where is a nilsequence as above, is small in norm, and is very small in norm; has small inner product with , , and , and thus with itself, and so and are both small in , giving the claim.)
For each , let denote the set of all such that there exists a degree nilsequence (depending on ) of complexity such that
From the Hardy-Littlewood maximal inequality (and the measure-preserving nature of ) we see that has measure . This implies that the functions
are uniformly bounded in as , which by Fatou’s lemma implies that
is also absolutely integrable. In particular, for almost every , we have
for some finite , which implies that
for an infinite sequence of (the exact choice of sequence depends on ); in particular, there is a such that for all in this sequence, one has
for all and all . Thus
for all in this sequence, all , and all ; combining with (2) we see (for almost every ) that
and thus for all , all , and all we have
where the limit is along the sequence.
For given , there are only finitely many possibilities for the nilmanifold , so by the usual diagonalisation argument we may pass to a subsequence of and assume that does not depend on for any . By Arzela-Ascoli we may similarly assume that the Lipschitz function converges uniformly to , so we now have
along the remaining subsequence for all , all , and all .
By repeatedly breaking the coefficients of the polynomial sequence into fractional parts and integer parts, and absorbing the latter in , we may assume that these coefficients are bounded. Thus, by Bolzano-Weierstrass and refining the sequence of further, we may assume that converges locally uniformly in as goes to infinity to a polynomial sequence , for every . We thus have (for almost every ) that
for all , all , and all . Henceforth we shall cease to keep control of the complexity of or .
for all , all , and some . By replacing various nilsystems with Cartesian powers, we may assume that the nilsystems are increasing in and in the sense that the nilsystem for is a factor of that for or , with the origin mapping to the origin. Then, by restricting to the orbit of the origin, we may assume that all the nilsystems are ergodic (and thus also uniquely ergodic, by the special properties of nilsystems). The nilsystems then have an ergodic inverse limit with an origin , and each function lifts up to a continuous function on , with . Thus
From the triangle inequality we see in particular that
for all and all , which by unique ergodicity of the nilsystems implies that
Thus the sequence is Cauchy in and tends to a some limit .
If is generic for (which is true for almost every ), we conclude from (4) and unique ergodicity of nilsystems that
for , which on taking limits as gives
A similar argument gives (1) for almost every , for each choice of . Since one only needs to verify a countable number of these conditions, we can find an for which all the (1) hold simultaneously, and the claim follows.
Remark 6 In order to use the combinatorial inverse theorem to prove the full ergodic inverse theorem (and not just the weak version), it appears that one needs an “algorithmic” or “measurable” version of the combinatorial inverse theorem, in which the nilsequence produced by the inverse theorem can be generated in a suitable “algorithmic” sense from the original function . In the setting of the inverse theorem over finite fields, a result in this direction was established by Tulsiani and Wolf (building upon a well-known paper of Goldreich and Levin handling the case). It is thus reasonable to expect that a similarly algorithmic version of the combinatorial inverse conjecture is true for higher Gowers uniformity norms, though this has not yet been achieved in the literature to my knowledge.