Multiple ergodic averages for variable polynomials

The study of multiple ergodic averages along polynomials dates back to 1977. Furstenberg, exploiting the $L^{2}$ limiting behavior (all the limits in this article are taken with respect to the $L^{2}$ norm, unless otherwise stated), as $N\to\infty,$ of

where $\ell\in\mathbb{N},$ $(X,\mathcal{B},\mu,T)$ is a measure preserving system,¹¹1 I.e., $T:X\to X$ is an invertible measure preserving transformation on a standard Borel probability space $(X,\mathcal{B},\mu).$ and $f_{1},\ldots,f_{\ell}\in L^{\infty}(\mu),$ provided (in [16]) a purely ergodic theoretic proof of Szemerédi’s theorem; every subset of natural numbers of positive upper density²²2For a set $A\subseteq\mathbb{N}$ we define its upper density, $\bar{d}(A),$ as $\bar{d}(A):=\limsup_{N}N^{-1}\cdot|A\cap\{1,\ldots,N\}|.$ contains arbitrarily long arithmetic progressions (a result that can be immediately obtained by combining Theorem 2.3 with Theorem 2.4 below).

A polynomial $p\in\mathbb{Q}[t]$ is an integer polynomial if $p(\mathbb{Z})\subseteq\mathbb{Z}.$ It was Bergelson who first visualized the iterates $n,2n,\ldots,\ell n$ in (1) as linear “distinct enough” integer polynomials. The integer polynomials $p_{1},\ldots,p_{\ell},$ $\ell\in\mathbb{N},$ are essentially distinct if $p_{i},$ $p_{i}-p_{j}$ are non-constant for all $i\neq j.$ Bergelson studied (initially in [2]), via the use of van der Corput’s lemma, a crucial tool in “reducing the complexity” of the iterates, averages of the form

for essentially distinct integer polynomials $p_{1},\ldots,p_{\ell}$; this study eventually led to multidimensional polynomial extensions of Szemerédi’s theorem (see [5]).

Bergelson and Leibman conjectured (in [4]) that multiple ergodic averages of the form

in any system, for multiple commuting $T_{i}$’s (i.e., $T_{i}T_{j}=T_{j}T_{i}$ for all $i,j$) and arbitrary integer polynomials $p_{i}$, always have limit (as $N\to\infty$). This conjecture was answered in the positive by Walsh, who actually showed it in greater generality (see [27]). No specific expression of the limit was provided by the method.³³3 The conjecture corresponding to that of Bergelson and Leibman about iterates which are integer parts of real polynomials, was shown in [23].

One of the questions that someone is called upon to answer is under which conditions, either on the polynomials or the system, we can explicitly find the limit of the aforementioned expressions. In particular, whether we can find families of polynomials for which we have convergence in a general system to a specific expression (see more about the “expected” limit below); then we can get a number of interesting applications, e.g., find the corresponding arithmetic configurations on “large” subsets of integers. For instance, showing that the characteristic factor coincides with the nilfactor of the system, and exploiting the equidistribution property of the corresponding polynomial sequence in nilmanifolds (all these notions will be defined later), Frantzikinakis proved (in [12]) that the expression

where $p\in\mathbb{R}[t]$ with $p(t)\neq cq(t)+d,$ $c,d\in\mathbb{R},$ $q\in\mathbb{Q}[t],$ has the same limit (as $N\to\infty$), in any system, as (1); obtaining a refinement of Szemerédi’s theorem.⁴⁴4 Such polynomials $p$ have the property that for every $\lambda\in\mathbb{R}\setminus\{0\},$ $\lambda p$ has at least one non-constant irrational coefficient, which is exactly the case (via Weyl’s criterion) when the corresponding sequence $(\lambda p(n))_{n}$ is equidistributed.

Generalizing the condition to multiple polynomials, following Frantzikinakis’ approach, Karageorgos and the author showed (in [20]) that for strongly independent real polynomials $p_{1},\ldots,p_{\ell}$ (i.e., any non-trivial linear combination of the $p_{i}$’s with scalars from $\mathbb{R}$ has at least one non-constant irrational coefficient) the expression

has the “expected” limit. In order to explain what is meant by “expected” limit, we need to recall the ergodicity and weakly mixing notions. $T$ is ergodic if $T^{-1}A=A$ implies $\mu(A)\in\{0,1\}$; $T$ is weakly mixing if $T\times T$ is ergodic. Here, by “expected” limit we mean, in case $T$ is ergodic, that the limit is equal to $\prod_{i=1}^{\ell}\int f_{i}\;d\mu,$ whereas, in the general case, it is equal to $\prod_{i=1}^{\ell}\mathbb{E}(f_{i}|\mathcal{I}(T)),$ where $\mathbb{E}(f_{i}|\mathcal{I}(T))$ is the conditional expectation of $f_{i}$ with respect to the $\sigma$-algebra of the $T$-invariant sets (notice here the connection to independence in probability). Furstenberg showed in [16] that for a weakly mixing $T,$ (1) converges to the expected limit; under the same assumption on $T$, Bergelson showed in [2] that (2) converges to the same limit as well.

We extend the distinctness property of the sequences of iterates of (5) to sequences of real variable polynomials:

For the class of good polynomial sequences, Frantzikinakis stated the following problem:

As stated in [13], Problem 1 is interesting even in the special cases of Examples 1 and  2.

Showing that (4) has the same limit as (1) for $p\in\mathbb{R}[t]$ with $p(t)\neq cq(t)+d,$ $c,d\in\mathbb{R},$ $q\in\mathbb{Q}[t],$ which follows from [12, Theorem 2.2], one comes to the following problem, which is a natural generalization of Frantzikinakis’ result to good-variable-polynomials:

As mentioned in [13], the $\ell=1$ case of Problem 1 (which also coincides with the $\ell=1$ case of Problem 2) can be easily obtained by using the spectral theorem. For general $\ell\in\mathbb{N},$ we make progress towards the solution of both Problems 1 and 2. In particular, under some additional assumptions on the coefficients of the good variable polynomials, we show two general results: Theorems 2.1 and  2.2. In this introductory section, we will present an easier application of each of them, which still covers both Examples 1 and  2.

To this end, we first recall the set of sublinear logarithmico-exponential Hardy field functions (of polynomial degree $0$) which converge (as $x\to\infty$) to ($\pm$) infinity:⁵⁵5 Let $R$ be the collection of equivalence classes of real valued functions defined on some halfline $(c,\infty),$ $c\geq 0,$ where two functions that agree eventually are identified. These classes are called germs of functions. A Hardy field is a subfield of the ring $(R,+,\cdot)$ that is closed under differentiation. Here, we use the word function when we refer to elements of $R$ (understanding that all the operations defined and statements made for elements of $R$ are considered only for sufficiently large $x\in\mathbb{R}$). We say that $g$ is a logarithmico-exponential Hardy field function, and we write $g\in\mathcal{LE},$ if it belongs to a Hardy field of real valued functions and it is defined on some $(c,+\infty),$ $c\geq 0,$ by a finite combination of symbols $+,-,\times,\div,\sqrt[n]{\cdot},\exp,\log$ acting on the real variable $x$ and on real constants. For more on Hardy field functions, see [10, 12, 18].

(we write $g_{2}\prec g_{1}\text{ if }|g_{1}(x)|/|g_{2}(x)|\to\infty\text{ as }x\to\infty$). Next, we define an appropriate set of coefficients: For $g_{i}\in\mathcal{SLE},$ $1\leq i\leq l,$ $l\in\mathbb{N},$ with $g_{l}\prec\ldots\prec g_{1},$⁶⁶6 The different growth relation between the $g_{i}$’s, $1\leq i\leq l,$ is postulated to avoid cases as, e.g., $(N+1)^{-1/2}-N^{-1/2}\sim N^{-3/2},$ since $g(x)=x^{3/2}$ is not sublinear (here we write $g_{2}\sim g_{1}$ if $g_{1}(x)/g_{2}(x)\to c\in\mathbb{R}\setminus\{0\}$). let $\mathcal{C}(g_{1},\ldots,g_{l})$ be the set of all linear combinations of reciprocals of the $g_{i}$’s, i.e.,

Extending the definition from [20], we say that the sequence of $\ell$-tuple of variable polynomials $(p_{1,N},\ldots,p_{\ell,N})_{N},$ where for each $1\leq i\leq\ell,$ $p_{i,N}$ has the form:

is strongly independent if for any $(\lambda_{1},\ldots,\lambda_{\ell})\in\mathbb{R}^{\ell}\setminus\{\vec{0}\}$ we have that $\sum_{i=1}^{\ell}\lambda_{i}p_{i,N}(n)$ is a non-constant polynomial in $n$. For example, the following triple of variable polynomials is strongly independent:

where $g_{5}(x):=\log x\prec g_{4}(x):=\log x\cdot\log\log x\prec g_{3}(x):=x^{1/2}\prec g_{2}(x):=x^{\pi/4}\log^{3/2}x\prec g_{1}(x):=x^{e/3}.$⁷⁷7 If $p_{1,N}(n):=\Big{(}\frac{\sqrt{2}}{g_{3}(N)}+\frac{1}{g_{4}(N)}\Big{)}n^{3}-\frac{31}{g_{5}(N)}n+1,$ $p_{2,N}(n):=\frac{1}{g_{3}(N)}n^{3},$ and $p_{3,N}(n):=\Big{(}\frac{\sqrt{3}}{g_{1}(N)}-\frac{17}{g_{2}(N)}\Big{)}n^{2},$ we have that $\lambda_{1}p_{1,N}+\lambda_{2}p_{2,N}+\lambda_{3}p_{3,N}$ is constant only when $\lambda_{1}=\lambda_{2}=\lambda_{3}=0.$ Regarding Problem 1, i.e., multiple variable polynomial sequences, we have the following result:

Regarding Problem 2, our result is the following theorem:

Very few convergence results for averages with polynomial iterates, in which we can explicitly find the limit, exist. Results for variable polynomials are even scarcer. We will conclude this introduction, by mentioning some of them. Kifer ([21]) studied multiple averages for variable polynomials of the form $p_{i,N}(n)=p_{i}(n)+q_{i}(N),$ with $p_{i}$’s essentially distinct and $q_{i}(\mathbb{Z})\subseteq\mathbb{Z}$ for a weakly mixing transformation $T.$ Similarly, for more general polynomials, Kifer studied averages for strongly mixing “enough” transformations. Finally, Frantzikinakis (in [12]) found characteristic factors (see Definition 4.1) for averages with variable polynomial iterates $p_{i,N},$ with leading coefficients independent of $N$. It is the arguments from this article ([12]) that we will adapt, in order to find characteristic factors for the averages appearing in Theorems  1.2 and  1.3 as well, which is one of the main two ingredients of the proof (the second one is the equidistribution of particular sequences for which we adapt arguments from [11]).

Here we will state our most general results and some applications. For the proofs, we follow [10] and [20], adapting the corresponding arguments to the variable polynomial case.

We first cover Problem 1 for a subclass of good polynomial sequences:

We also cover the following case of Problem 2:

We will show that Theorem 2.1 implies Theorem 1.2 (resp. Theorem 2.2 implies Theorem 1.3), and that it holds for any polynomial family $\{p_{1},\ldots,p_{\ell}\}$ (resp. $\{p,2p,\ldots,\ell p\}$ for Theorem 2.2) which is independent of $N$ and for which non-trivial linear combinations of its members satisfy (6). In particular, it generalizes [20, Theorem 2.1] for strongly independent polynomials (the same is true for Theorem 2.2 for the single polynomial case).

The approach we follow to show these results is similar to the one in [12, 20], with a few extra twists. Namely, one has to find the characteristic factors of (8) and (9), and show some equidistribution results in nilmanifolds. The “super niceness” property (Definition 4.8) will be introduced so we can deal with the former, while the “goodness” property (Definition 1.1) implies the latter.

As was mentioned in the previous section, the ergodicity assumption in Theorem 2.1 can be dropped.⁹⁹9 The limit in this case is equal to $\prod_{i=1}^{\ell}\mathbb{E}(f_{i}|\mathcal{I}(T)).$ Indeed, if $\mu=\int\mu_{t}\;d\lambda(t)$ denotes the ergodic decomposition of $\mu,$ it suffices to show that if $\mathbb{E}(f_{i}|\mathcal{I}(T))=0$ for some $i,$ then the averages converge to $0.$ Since $\mathbb{E}(f_{i}|\mathcal{I}(T))=0,$ we have that $\int f_{i}\;d\mu_{t}=0$ for $\lambda$-a.e. $t.$ By (8), we have that the averages go to $0$ in $L^{2}(\mu_{t})$ for $\lambda$-a.e. $t,$ hence the limit is equal to $0$ in $L^{2}(\mu)$ by the Dominated Convergence Theorem. Hence, the theorems hold for any system; their strong nature is also reflected in the fact that they have immediate recurrence and combinatorial implications which we discuss next.