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In the modern theory of higher order Fourier analysis, a key role are played by the Gowers uniformity norms for
. For finitely supported functions
, one can define the (non-normalised) Gowers norm
by the formula
The significance of the Gowers norms is that they control other multilinear forms that show up in additive combinatorics. Given any polynomials and functions
, we define the multilinear form
-
and
have true complexity
;
-
has true complexity
;
-
has true complexity
;
- The form
(which among other things could be used to count twin primes) has infinite true complexity (which is quite unfortunate for applications).
Gowers and Wolf formulated a conjecture on what this complexity should be, at least for linear polynomials ; Ben Green and I thought we had resolved this conjecture back in 2010, though it turned out there was a subtle gap in our arguments and we were only able to resolve the conjecture in a partial range of cases. However, the full conjecture was recently resolved by Daniel Altman.
The (semi-)norm is so weak that it barely controls any averages at all. For instance the average
Because of this, I propose inserting an additional norm in the Gowers uniformity norm hierarchy between the and
norms, which I will call the
(or “profinite
“) norm:
The norm recently appeared implicitly in work of Peluse and Prendiville, who showed that the form
had true complexity
in this notation (with polynomially strong bounds). [Actually, strictly speaking this control was only shown for the third function
; for the first two functions
one needs to localize the
norm to intervals of length
. But I will ignore this technical point to keep the exposition simple.] The weaker claim that
has true complexity
is substantially easier to prove (one can apply the circle method together with Gauss sum estimates).
The well known inverse theorem for the norm tells us that if a
-bounded function
has
norm at least
for some
, then there is a Fourier phase
such that
For one has a trivial inverse theorem; by definition, the
norm of
is at least
if and only if
For one has the intermediate situation in which the frequency
is not taken to be zero, but is instead major arc. Indeed, suppose that
is
-bounded with
, thus
Here is a diagram showing some of the control relationships between various Gowers norms, multilinear forms, and duals of classes of functions (where each class of functions
induces a dual norm
:
Here I have included the three classes of functions that one can choose from for the inverse theorem, namely degree two nilsequences, bracket quadratic phases, and local quadratic phases, as well as the more narrow class of globally quadratic phases.
The Gowers norms have counterparts for measure-preserving systems , known as Host-Kra seminorms. The
norm can be defined for
as
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.
In this lecture, we describe the simple but fundamental Furstenberg correspondence principle which connects the “soft analysis” subject of ergodic theory (in particular, recurrence theorems) with the “hard analysis” subject of combinatorial number theory (or more generally with results of “density Ramsey theory” type). Rather than try to set up the most general and abstract version of this principle, we shall instead study the canonical example of this principle in action, namely the equating of the Furstenberg multiple recurrence theorem with Szemerédi’s theorem on arithmetic progressions.
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