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 that
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:
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 has
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:
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 has
for all non-zero
, where
. Then
is a factor of 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.
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
for all and
, and all polynomials
. Of course it suffices to do so for those polynomials with rational coefficients (so now there are only a countable number of constraints to consider).
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
then has measure at most
(say), and so the function
is absolutely integrable. By the maximal ergodic theorem, we thus see that for almost every
, there exists a finite
such that
for all and all
. Informally, we thus have the approximation
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
.
We can lift the polynomial sequence up to a linear sequence (enlarging as necessary), thus
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.
5 comments
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23 July, 2015 at 7:57 pm
Maurice Nivat
do you have any thing to say about the décomposition of a set of integers A into the Minkowski sum of two other sets B and C, non trivial if card B and card C aree both at least equal to 2 I raised the question on research gate can the décomposability of A be checked in poly time, either polynoial in card A or plynomiel in l(A)= maxA-minA? regards Maurie Nivat
24 July, 2015 at 9:49 am
Deducing the inverse theorem for the multidimensional Gowers norms from the one-dimensional version | What's new
[…] mentioned in the previous two posts, Ben Green, Tamar Ziegler, and myself proved the following inverse theorem for the Gowers […]
25 July, 2015 at 3:24 pm
Schlafly
You are in the NY Times, Terry! Please make a separate post for that, so your loyal fans can comment on it without polluting your discussion of nilsequences and more important topics.
29 September, 2015 at 1:45 pm
gregdreads
A little math is nice but damn!
27 March, 2016 at 5:18 pm
Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteristic factors; polynomial patterns in primes | What's new
[…] 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 […]