Andreas Koutsogiannis, Anh Le, Florian Richter and I have just uploaded to the arXiv a new paper entitled “Structure of multicorrelation sequences with integer part polynomial iterates along primes”. This short paper establishes a natural common extension of our earlier result with Le and Richter and a theorem of Koutsogiannis. I decided this was a good opportunity to write more generally about multicorrelation sequences and their relation to nilsequences.
— 1. Single transformation —
Here, as usual, we denote by the function . This is a natural extension of the concept of single correlation sequences, which correspond to the case and control, for instance, spectral and mixing properties of the system. Multicorrelation sequences (or multiple correlation sequences) appear naturally in the theory of multiple recurrence. For instance, if for some with , then the fact that for some is the statement of Furstenberg’s famous multiple recurrence theorem, which, via the correspondence principle, is equivalent to the celebrated Szemerédi theorem on arithmetic progressions.
— 1.1. The Bergelson-Host-Kra theorem —
It turns out that, despite the large number of degrees of freedom involved, multicorrelation sequences are quite rigid. This was first captured in a beautiful result of Bergelson Host and Kra in 2005, which shows that multicorrelation sequences are always only a small perturbation away from a very structured kind of sequence they named nilsequence. Since their debut in the aforementioned paper, nilsequences have became surprisingly important in additive combinatorics and number theory, due largely to the Inverse Theorem for Gowers norms.
There are several variants of the notion of nilsequence. In this post I chose to use the definition given in the original paper of Bergelson Host and Kra (in our paper we use the more common terminology to call a nilsequence to what I defined above as a basic nilsequence). In some other variants is allowed different degrees of regularity (Riemann integrability, or Lipschitz or even ).
It will also be convenient to use the Weyl and the Besicovitch seminorms on , defined for , respectively, by the formulas
— 1.2. Averaging along subsequences —
Several enhancements of Szemerédi’s theorem involve restrictions on the common difference of the arithmetic progression to be found inside an arbitrary set with positive density. This leads to the study of averages of multicorrelation sequences along subsequences of natural numbers. It turns out that the structure of multicorrelation sequences survives even when we restrict attention to moderately sparse subsequences. For instance, given defined by (1) and letting be given by Theorem 2, we have also that
One can write the previous equation as where . In fact, the same is true when is any polynomial with integer coefficients (see Theorem 3 below).
On a related direction, when is the increasing enumeration of the prime numbers, or when for an arbitrary (where denotes the floor function), then Le proved that . In this case one can not replace the Besicovitch seminorm with the Weyl seminorm.
Back to polynomials, a far reaching extension of Theorem 1 was obtained by Leibman in two papers published in 2010 and 2015. In particular, Leibman was able to remove the hypothesis that the system defining the multicorrelation sequence is ergodic.
By taking , we see that the class of multicorrelations defined by (1) is contained in the class defined by (2). On the other hand, multicorrelations of the form (2) appear naturally when considering polynomial extensions of Szemerédi’s theorem or of Furstenberg’s multiple recurrence theorem.
Part of the aforementioned result of Le also applies to multicorrelations of the form (2).
— 2. Commuting transformations —
Multidimensional extensions of Szemerédi’s theorem (which in turn are a significant special case of the density Hales-Jewett theorem) can also be obtained via ergodic theoretic methods but require the use of measure preserving systems with several commuting transformations. This in turn leads to a more general class of multicorrelation sequences:
where are commuting measure preserving transformations on the probability space and . Letting we see that the class of multicorrelation sequences defined by (3) contains the class of sequences defined by (1). In view of Theorem 2, the following is a natural conjecture.
Unfortunately, the proofs of the results mentioned in the previous section all rely on the structure theory of Host and Kra, which allows one to understand multiple ergodic averages by studying characteristic nilfactors, but which is unavailable for commuting transformations. In particular, this conjecture is still open. Nevertheless, in 2014, Frantzikinakis was able to obtain the following
A convenient way to encode a tuple of commuting measure preserving transformations on a probability space is via a single measure preserving action of , defined by , where is the -th vector in the canonical basis of . Given a measure preserving -action on a probability space and vector polynomials (i.e. each for some ) one can consider the multicorrelation sequence
The class of multicorrelation sequences defined by (4) contains all the other classes of multicorrelation sequences mentioned before in this post and appear naturally when studying multidimensional polynomial extensions of Szemerédi’s theorem. Theorem 6 was proved by Frantzikinakis also in the more general situation where is given by (4).
— 3. Integer part of real polynomials —
Given a measure preserving -action on a probability space , vector polynomials (i.e., each with ) and functions , one can construct the multicorrelation sequence
where for a vector we denote . If all polynomials have only integer coefficients, then (5) becomes (4), so the class of multicorrelation sequences defined via (5) contains all the other multicorrelation sequences defined in this post.
The extension of Frantzikinakis theorem to multicorrelation sequences of the form (5) was obtained by Koutsogiannis.
— 4. Proofs —
As already mentioned above, the proofs of the results listed in Section 1 all require the structure theory. This involves two steps: first is a reduction to nilsystems and then an analysis of the equidistribution properties of orbits in nilsystems. I plan to write another post about these ideas in the near future.
The breakthrough idea of Frantzikinakis, leading to his proof of Theorem 6 can be described in rough terms as follows. Using a common van der Corput-type procedure, one can show that a multicorrelation sequence of the form (3) is orthogonal, with respect to the seminorm, to any sequence which is uniform (in a sense analogous to Gowers uniformity in finite groups.) The same is true for any nilsequence. Then, letting be the orthogonal projection of onto the space of all nilsequences (w.r.t. ) it follows that the difference is also orthogonal to every uniform sequence . By (a version of) the inverse theorem for uniformity seminorms, if were not negligible, then it would have to correlate with some nilsequence. This is not possible, since is the orthogonal projection of to the space of nilsequences, and it follows that . Unfortunately, the space of nilsequences is not closed in the seminorm, so itself is not a nilsequence. Nevertheless it belongs to the -closure of the space of nilsequences, so for every there is a nilsequence such that .
When proving Theorem 7, because the primes are involved, we need to employ the -trick. This in turn requires some control on the nilsequences that we pick. In order to obtain the necessary additional properties on we have to first improve Theorem 6. We do this by describing the (topological) Furstenberg system of a multicorrelation sequence .
Finally, the proof of Theorem 9 follows closely the proof of Theorem 8. The idea is to approximate as defined in (5) by a multicorrelation sequence as defined in (4). To do this we pass to (a multidimensional analogue of) the suspension flow and use the fact that, if is a -action and then
Since the measure preserving transformations commute, by extending the original -action to a -action one can represent a real polynomial by polynomials with integer coefficients. There are of course some difficulties with implementing this strategy, since one also needs to handle the floor function, but this is ultimately taken care of by the fact that polynomials, and polynomials evaluated along primes, equidistribute mod in a well understood subset of the interval.