Structure of multicorrelation sequences with integer part polynomial iterates along primes

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 —

Given an invertible measure preserving system ${(X,\mu,T)}$ and bounded functions ${f_0,\dots,f_k\in L^\infty(X)}$ one can form the multicorrelation sequence

$\displaystyle \alpha(n)=\int_Xf_0\cdot T^n f_1\cdots T^{kn}f_k\,\mathrm{d}\mu. \ \ \ \ \ (1)$

Here, as usual, we denote by ${T^nf}$ the function ${f\circ T^n}$. This is a natural extension of the concept of single correlation sequences, which correspond to the case ${k=1}$ 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 ${f_0=f_1=\cdots=f_k=1_A}$ for some ${A\subset X}$ with ${\mu(A)>0}$, then the fact that ${\alpha(n)>0}$ for some ${n}$ 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.

Definition 1 Let ${G}$ be a nilpotent Lie group and let ${\Gamma\subset G}$ be a discrete and co-compact subgroup. The (compact) homogeneous space ${X=G/\Gamma}$ is called a nilmanifold and ${G}$ acts naturally on ${X}$ via left multiplication. Given a continuous function ${F:X\rightarrow{\mathbb C}}$, a point ${x\in X}$ and ${g\in G}$, the sequence ${\phi:{\mathbb N}\rightarrow{\mathbb C}}$ defined via ${\phi(n)=F(g^nx)}$ is called a basic nilsequence. A sequence ${\psi}$ is called a nilsequence if for every ${\epsilon>0}$ there exists a basic nilsequence ${\phi}$ such that ${\|\phi-\psi\|_{\ell^\infty}<\epsilon}$.

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 ${F}$ is allowed different degrees of regularity (Riemann integrability, or Lipschitz or even ${C^\infty}$).

It will also be convenient to use the Weyl and the Besicovitch seminorms on ${\ell^\infty}$, defined for ${f\in\ell^\infty}$, respectively, by the formulas

$\displaystyle \|f\|_W:=\limsup_{N-M\rightarrow\infty}\frac1{N-M}\sum_{n=M}^N\big|f(n)\big|\qquad \|f\|_B:=\limsup_{N\rightarrow\infty}\frac1{N}\sum_{n=1}^N\big|f(n)\big|$

Theorem 2 (Bergelson-Host-Kra, 2005) Let ${\alpha}$ be a multicorrelation sequence defined by (1) where ${(X,\mu,T)}$ is an ergodic system. Then there exists a nilsequence ${\psi}$ such that ${\|\alpha-\psi\|_W=0}$.

— 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 ${\alpha}$ defined by (1) and letting ${\psi}$ be given by Theorem 2, we have also that

$\displaystyle \lim_{N-M\rightarrow\infty}\frac1{N-M}\sum_{n=M}^N\big|\alpha(n^2)-\psi(n^2)\big|=0.$

One can write the previous equation as ${\|(\alpha-\psi)\circ p\|_W=0}$ where ${p(n)=n^2}$. In fact, the same is true when ${p}$ is any polynomial with integer coefficients (see Theorem 3 below).

On a related direction, when ${p(n)}$ is the increasing enumeration of the prime numbers, or when ${p(n)=\lfloor n^c\rfloor}$ for an arbitrary ${c\in{\mathbb R}^{>0}}$ (where ${\lfloor\cdot\rfloor}$ denotes the floor function), then Le proved that ${\|(\alpha-\psi)\circ p\|_B=0}$. 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.

Theorem 3 (Leibman, 2010-2015) Let ${(X,\mu,T)}$ be a measure preserving system and let ${f_0,\dots,f_k\in L^\infty(X)}$. Let ${p_1,\dots,p_k\in{\mathbb Z}[x]}$ and let

$\displaystyle \alpha(n)=\int_Xf_0\cdot T^{p_1(n)} f_1\cdots T^{p_k(n)}f_k\,\mathrm{d}\mu. \ \ \ \ \ (2)$

Then there exists a nilsequence ${\psi}$ such that ${\|\alpha-\psi\|_W=0}$.

By taking ${p_i(n)=in}$, 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 ${\alpha}$ of the form (2).

Theorem 4 (Le, 2017) Let ${\alpha}$ be defined by (2), let ${\psi}$ be the nilsequence given by Theorem 3 and let ${p(n)}$ be the increasing enumeration of the prime numbers. Then ${\big\|(\alpha-\psi)\circ p\big\|_B=0}$.

— 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:

$\displaystyle \alpha(n)=\int_Xf_0\cdot T_1^n f_1\cdots T_k^nf_k\,\mathrm{d}\mu, \ \ \ \ \ (3)$

where ${T_1,\dots,T_\ell}$ are commuting measure preserving transformations on the probability space ${(X,\mu,T)}$ and ${f_0,\dots,f_k\in L^\infty(X)}$. Letting ${T_i=T^i}$ 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.

Conjecture 5 Let ${\alpha}$ be defined by (3). Does there exist a nilsequence ${\psi}$ such that ${\|\alpha-\psi\|_W=0}$?

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

Theorem 6 (Frantzikinakis) Let ${\alpha}$ be defined by (3). Then for every ${\epsilon>0}$ there exists a nilsequence ${\psi}$ such that ${\|\alpha-\psi\|_W<\epsilon}$.

Remark 1 Since ${\|a\|_W\leq\|a\|_{\ell^\infty}}$ for any ${a\in\ell^\infty}$, in Theorem 6 one can require ${\psi}$ to be basic nilsequence. To streamline the presentation I will keep the slightly weaker formulation in this and other results below.

A convenient way to encode a tuple of ${\ell}$ commuting measure preserving transformations ${T_1,\dots,T_\ell}$ on a probability space ${(X,\mu)}$ is via a single measure preserving action ${T=(T^v)_{v\in{\mathbb Z}^\ell}}$ of ${{\mathbb Z}^\ell}$, defined by ${T^{e_i}=T_i}$, where ${e_i}$ is the ${i}$-th vector in the canonical basis of ${{\mathbb Z}^\ell}$. Given a measure preserving ${{\mathbb Z}^\ell}$-action ${T}$ on a probability space and vector polynomials ${p_1,\dots,p_k:{\mathbb Z}\rightarrow{\mathbb Z}^\ell}$ (i.e. each ${p_i=(p_{i,1},\dots,p_{i,\ell})}$ for some ${p_{i,j}\in{\mathbb Z}[x]}$) one can consider the multicorrelation sequence

$\displaystyle \alpha(n)=\int_Xf_0\cdot T^{p_1(n)} f_1\cdots T^{p_k(n)} f_k\,\mathrm{d}\mu. \ \ \ \ \ (4)$

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 ${\alpha}$ is given by (4).

Earlier this year, Le, Richter and I obtained a refinement of Theorem 6, which stands to it in the same way that Theorem 4 stands to Theorem 2.

Theorem 7 Let ${\alpha}$ be defined by (4) and let ${p(n)}$ be the increasing enumeration of the prime numbers. Then for every ${\epsilon>0}$ there exists a nilsequence ${\psi}$ satisfying simultaneously ${\|\alpha-\psi\|_W<\epsilon}$ and ${\big\|(\alpha-\psi)\circ p\|_B<\epsilon}$.

— 3. Integer part of real polynomials —

Given a measure preserving ${{\mathbb Z}^\ell}$-action ${T}$ on a probability space ${(X,\mu)}$, vector polynomials ${p_1,\dots,p_k:{\mathbb R}\rightarrow{\mathbb R}^\ell}$ (i.e., each ${p_i=(p_{i,1},\dots,p_{i,\ell})}$ with ${p_{i,j}\in{\mathbb R}[x]}$) and functions ${f_0,\dots,f_k\in L^\infty(X)}$, one can construct the multicorrelation sequence

$\displaystyle \alpha(n)=\int_Xf_0\cdot T^{\lfloor p_1(n)\rfloor} f_1\cdots T^{\lfloor p_k(n)\rfloor} f_k\,\mathrm{d}\mu, \ \ \ \ \ (5)$

where for a vector ${x=(x_1,\dots,x_\ell)\in{\mathbb R}^\ell}$ we denote ${\lfloor x\rfloor=(\lfloor x_1\rfloor,\dots,\lfloor x_\ell\rfloor)}$. If all polynomials ${p_i}$ 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.

Theorem 8 (Koutsogiannis) Let ${\alpha}$ be defined by (5). Then for every ${\epsilon>0}$ there exists a nilsequence ${\psi}$ such that ${\|\alpha-\psi\|_W<\epsilon}$.

The main result of our paper is the common extension of Theorems 7 and 8.

Theorem 9 Let ${\alpha}$ be defined by (5) and let ${p(n)}$ be the increasing enumeration of the prime numbers. Then for every ${\epsilon>0}$ there exists a nilsequence ${\psi}$ satisfying simultaneously ${\|\alpha-\psi\|_W<\epsilon}$ and ${\big\|(\alpha-\psi)\circ p\|_B<\epsilon}$.

— 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 ${\alpha}$ of the form (3) is orthogonal, with respect to the ${\|\cdot\|_W}$ seminorm, to any sequence ${\beta\in\ell^\infty}$ which is uniform (in a sense analogous to Gowers uniformity in finite groups.) The same is true for any nilsequence. Then, letting ${\gamma}$ be the orthogonal projection of ${\alpha}$ onto the space of all nilsequences (w.r.t. ${\|\cdot\|_W}$) it follows that the difference ${\alpha-\gamma}$ is also orthogonal to every uniform sequence ${\beta}$. By (a version of) the inverse theorem for uniformity seminorms, if ${\alpha-\gamma}$ were not negligible, then it would have to correlate with some nilsequence. This is not possible, since ${\gamma}$ is the orthogonal projection of ${\alpha}$ to the space of nilsequences, and it follows that ${\|\alpha-\gamma\|_W=0}$. Unfortunately, the space of nilsequences is not closed in the ${\|\cdot\|_W}$ seminorm, so ${\gamma}$ itself is not a nilsequence. Nevertheless it belongs to the ${\|\cdot\|_W}$-closure of the space of nilsequences, so for every ${\epsilon>0}$ there is a nilsequence ${\psi}$ such that ${\|\alpha-\psi\|_W<\epsilon}$.

When proving Theorem 7, because the primes are involved, we need to employ the ${W}$-trick. This in turn requires some control on the nilsequences ${\psi}$ that we pick. In order to obtain the necessary additional properties on ${\psi}$ we have to first improve Theorem 6. We do this by describing the (topological) Furstenberg system of a multicorrelation sequence ${\alpha}$.

Finally, the proof of Theorem 9 follows closely the proof of Theorem 8. The idea is to approximate ${\alpha}$ 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 ${T}$ is a ${{\mathbb R}}$-action and ${q(n)=a_0+a_1n+\cdots+a_mn^m\in{\mathbb R}[n]}$ then

$\displaystyle T^{q(n)}=T^{a_0}\big(T^{a_1}\big)^n\big(T^{a_2}\big)^{n^2}\cdots\big(T^{a_m}\big)^{n^m}.$

Since the measure preserving transformations ${T^{a_1},\dots,T^{a_m}}$ commute, by extending the original ${{\mathbb Z}^\ell}$-action to a ${{\mathbb Z}^{\ell m}}$-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 ${1}$ in a well understood subset of the ${[0,1)}$ interval.

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