1.1 A brief history

In 1933, Khintchin [Reference Khintchin40] had the great insight to see how to generalize the classical equidistribution result of Bohl [Reference Bohl12], Sierpiński [Reference Sierpiński56] and Weyl [Reference Weyl66] from 1910 to a pointwise ergodic theorem, observing that as a consequence of Birkhoff’s famous ergodic theorem [Reference Birkhoff11], the following equidistribution result holds: namely, for any irrational $\theta \in {\mathbb R}$ , for any Lebesgue measurable set $E\subseteq [0,1)$ and for almost every $x\in {\mathbb R}$ ,

$$ \begin{align*} \lim_{M\to\infty} \frac{\# \{ m\in[M]: \{x + m \theta\} \in E \}}{M} = |E|, \end{align*} $$

where $\{x\}$ denotes the fractional part of $x\in \mathbb {R}$ , and $[N]:=(0, N]\cap {\mathbb {Z}}$ for any real number $N\ge 1$ . In 1916, Weyl [Reference Weyl67] extended the classical equidistribution theorem to general polynomial sequences $(\{P(n)\})_{n\in \mathbb {N}}$ having at least one irrational coefficient, and so it was natural to ask whether a pointwise ergodic extension of Weyl’s equidistribution theorem holds. This question was posed by Bellow [Reference Bellow5] and Furstenberg [Reference Furstenberg24] in the early 1980s; precisely, they asked if for any polynomial $P \in {\mathbb Z}[\mathrm {m}]$ with integer coefficients and $P(0)=0$ and for any invertible measure-preserving transformation $T: X \to X$ on a probability space $(X,\mathcal B(X), \mu )$ , does the limit

$$ \begin{align*} \lim_{M\to \infty} \mathbb{E}_{m\in[M]} f(T^{P(m)} x) \end{align*} $$

exist for almost every $x\in X$ and for every $f \in L^\infty (X)$ ? Here and throughout the paper we use the notation $\mathbb {E}_{y\in Y}f(y):=\frac {1}{\#Y}\sum _{y\in Y}f(y)$ for any finite set $Y\neq \emptyset $ and any function $f:Y\to \mathbb {C}$ . In the mid 1980s, the first author [Reference Bourgain13, Reference Bourgain14, Reference Bourgain15] established that this is indeed the case whenever $f \in L^p(X)$ and $p\in (1,\infty )$ , leaving open the question of what happens on $L^1(X)$ . Interestingly, it was shown much later by Buczolich and Mauldin [Reference Buczolich and Mauldin18] that the above pointwise convergence result fails for general $L^1$ functions when $P(m) = m^2$ ; see also [Reference LaVictoire42] for further refinements. In any case, the papers [Reference Bourgain13, Reference Bourgain14, Reference Bourgain15] represent a far-reaching common generalization of Birkhoff’s pointwise ergodic theorem and Weyl’s equidistribution theorem.

Both Birkhoff’s and Weyl’s results have natural multi-parameter extensions. In 1951, Dunford [Reference Dunford23] and Zygmund [Reference Zygmund72] independently extended Birkhoff’s theorem to multiple measure-preserving transformations $T_1, \ldots , T_k : X \to X$ . They showed that the limit

(1.1) $$ \begin{align} \lim_{M_1,\ldots,M_k \to \infty} \mathbb{E}_{(m_1,\ldots, m_k)\in\prod_{j=1}^k[M_j]} f(T_1^{m_1} \cdots T_k^{m_k} x) \end{align} $$

exists for almost every $x \in X$ and for any $f\in L^p(X)$ with $p\in (1, \infty )$ , where $\prod _{j=1}^k[M_j]:=[M_1]\times \ldots \times [M_k]$ . The limit is taken in the unrestricted sense; that is, when $\min \{M_1,\ldots ,M_k\} \to \infty $ . Here, when $k\ge 2$ , the pointwise convergence result is manifestly false for general $f\in L^1(X)$ .

In 1979, Arkhipov, Chubarikov and Karatsuba [Reference Arkhipov, Chubarikov and Karatsuba2] extended Weyl’s equidistribution result to polynomials (even multiple polynomials) of several variables. In its simplest form, their result asserts that for any k-variate polynomial $P\in {\mathbb {Z}}[\mathrm {m}_1,\ldots , \mathrm {m}_k]$ , any irrational $\theta \in \mathbb {R}$ and any interval $[a, b)\subseteq [0, 1)$ , one has

(1.2) $$ \begin{align} \lim_{\min\{M_1,\ldots,M_k\} \to \infty} \frac{\# \{(m_1, \ldots, m_k)\in\prod_{j=1}^k[M_j]: \{\theta P(m_1,\ldots, m_k)\} \in [a, b) \}}{M_1 \cdots M_k} = b-a. \end{align} $$

In the late 1980s, after [Reference Bourgain13, Reference Bourgain14, Reference Bourgain15] and in light of these results, it was natural to seek a common generalization of the results of Dunford and Zygmund on the one hand (which generalize Birkhoff’s original theorem) and Arkhipov, Chubarikov and Karatsuba on the other hand (which generalize Weyl’s theorem), which can be subsumed under the following conjecture, a multi-parameter variant of the Bellow–Furstenberg problem:

Conjecture 1.3. Let $k\in {\mathbb {Z}}_+$ with $k\ge 2$ be given and let $(X, \mathcal B(X), \mu )$ be a probability measure space with an invertible measure-preserving transformation $T:X\to X$ . Assume that $P\in {\mathbb {Z}}[\mathrm {m}_1, \ldots , \mathrm {m}_k]$ with $P(0)=0$ . Then for any $f\in L^{\infty }(X)$ , the limit

(1.4) $$ \begin{align} \lim_{\min\{M_1,\ldots,M_k\} \to \infty}\mathbb{E}_{(m_1,\ldots, m_k)\in\prod_{j=1}^k[M_j]} f(T^{P(m_1, \ldots, m_k)}x) \quad \text{exists for }\mu\text{-almost every }x\in X. \end{align} $$

Our main theorem resolves this conjecture.

The case $k=1$ corresponds to the classical one-parameter question of Bellow [Reference Bellow5] and Furstenberg [Reference Furstenberg24] and was resolved in [Reference Bourgain13, Reference Bourgain14, Reference Bourgain15]. In this paper, we will establish the cases $k\ge 2$ . In fact, we will prove stronger quantitative results including corresponding multi-parameter maximal and oscillation estimates (see Theorem 1.11 below), which will imply Conjecture 1.3. This paper also represents a first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems for ergodic averaging operators with polynomial orbits. Before we formulate our main quantitative results, we briefly describe the interesting history of Conjecture 1.3.

The theorems of Dunford [Reference Dunford23] and Zygmund [Reference Zygmund72] have simple proofs, which can be deduced by iterative applications of the classical Birkhoff ergodic theorem. For this purpose, it suffices to note that the Dunford–Zygmund averages from (1.1) can be written as a composition of k classical Birkhoff averages as follows

(1.6) $$ \begin{align} \mathbb{E}_{(m_1,\ldots, m_k)\in\prod_{j=1}^k[M_j]} f(T_1^{m_1} \cdots T_k^{m_k} x) = \mathbb{E}_{m_k\in[M_k]}\big[\cdots \mathbb{E}_{m_1\in[M_1]}f(T_1^{m_1}( \cdots T_k^{m_k}) x)\big]. \end{align} $$

The order in this composition is important since the transformations $T_1,\ldots , T_k$ do not need to commute. The first author, in view of [Reference Bourgain13, Reference Bourgain14, Reference Bourgain15], extended the observation from (1.6) to polynomial orbits and showed that for every $f\in L^p(X)$ with $p\in (1, \infty )$ , the limit

(1.7) $$ \begin{align} \lim_{\min\{M_1,\ldots,M_k\} \to \infty}\mathbb{E}_{(m_1,\ldots, m_k)\in\prod_{j=1}^k[M_j]} f(T_1^{P_1(m_1)} \cdots T_k^{P_k(m_k)} x) \end{align} $$

exists for $\mu $ -almost every $x\in X$ , whenever $P_1,\ldots , P_k\in {\mathbb {Z}}[\mathrm {m}]$ with $P_1(0)=\ldots =P_k(0)=0$ and $T_1,\ldots , T_d:X\to X$ is a family of commuting and invertible measure-preserving transformations. The result from (1.7) was never published; nonetheless, it can be thought of as a polynomial extension of the theorem of Dunford [Reference Dunford23] and Zygmund [Reference Zygmund72] (the arguments in Section 3.4 can be used to derive a quantitative version of (1.7)). Interestingly, as observed by Benjamin Weiss (privately communicated to the first author), any ergodic theorem for these averages fails in general for $k\ge 2$ when the $T_1,\ldots , T_k$ are general non-commuting transformations. It may even fail in the one-parameter situation for the averages of the form $\mathbb {E}_{m\in [M]} f(T_1^{P_1(m)} \cdots T_k^{P_k(m)} x)$ ; see also [Reference Bergelson and Leibman10] for interesting counterexamples.

This was a turning point, illustrating that the multi-parameter theory for averages with orbits along polynomials with separated variables as in (1.7) is well-understood and can be readily deduced from the one-parameter theory [Reference Bourgain13, Reference Bourgain14, Reference Bourgain15] by simple iteration as in (1.6). However, the equidistribution result (1.2) of Arkhipov, Chubarikov and Karatsuba [Reference Arkhipov, Chubarikov and Karatsuba2], based on the so-called multi-parameter circle method (deep and intricate tools in analytic number theory which go beyond the classical circle method) showed that the situation may be dramatically different when orbits are defined along genuinely k-variate polynomials $P\in {\mathbb {Z}}[\mathrm {m}_1, \ldots , \mathrm {m}_k]$ and led to Conjecture 1.3. Even for $k=2$ with $P(m_1, m_2)=m_1^2m_2^3$ in (1.4), the problem becomes very challenging. Surprisingly it seems that there is no simple way (like changing variables or interpreting the average from (1.4) as a composition of simpler one-parameter averages as in (1.6)) that would help us to reduce the matter to the setup where pointwise convergence is known.

The multi-parameter case $k\ge 2$ in Conjecture 1.3 lies in sharp contrast to the one-parameter situation $k=1$ , causing serious difficulties that were not apparent in [Reference Bourgain13, Reference Bourgain14, Reference Bourgain15]. The most notable differences are multi-parameter estimates of corresponding exponential sums and a delicate control of error terms that arise in implementing the circle method. These difficulties arise from the lack of nestedness when the parameters $M_1,\ldots ,M_k$ are independent; see Figure 1 and Figure 2 below. We now turn to a more detailed discussion and precise formulation of the results in this paper.

Figure 1 Family of nested rectangles (cubes) $Q_{M,M}\subset Q_{N,N}$ with $M<N$ , for $k=2$ .

Figure 2 Family of un-nested rectangles $Q_{M_1, M_2}\not \subseteq Q_{N_1, N_2}$ with $M_1<N_1$ and $M_2>N_2$ , for $k=2$ .

1.2 Statement of the main results

Throughout this paper, the triple $(X, \mathcal B(X), \mu )$ denotes a $\sigma $ -finite measure space, and ${\mathbb {Z}}[\mathrm {m}_1, \ldots , \mathrm {m}_k]$ denotes the space of all formal k-variate polynomials $P(\mathrm {m}_1, \ldots , \mathrm {m}_k)$ with $k\in {\mathbb {Z}}_+$ indeterminates $\mathrm {m}_1, \ldots , \mathrm {m}_k$ and integer coefficients. Each polynomial $P\in {\mathbb {Z}}[\mathrm {m}_1, \ldots , \mathrm {m}_k]$ will always be identified with a map ${\mathbb {Z}}^k\ni (m_1,\ldots , m_k)\mapsto P(m_1,\ldots , m_k)\in {\mathbb {Z}}$ .

Let $d, k \in {\mathbb {Z}}_+$ , and given a family ${\mathcal T} = \{T_1,\ldots , T_d\}$ of invertible commuting measure-preserving transformations on X, a measurable function f on X, polynomials ${\mathcal P} = \{P_1,\ldots , P_d\} \subset {\mathbb {Z}}[\mathrm m_1, \ldots , \mathrm {m}_k]$ and a vector of real numbers $M = (M_1,\ldots , M_k)$ whose entries are greater than $1$ , we define the multi-parameter polynomial ergodic average by

(1.8) $$ \begin{align} A_{{M}; X, {\mathcal T}}^{\mathcal P}f(x):= \mathbb{E}_{m\in Q_{M}}f(T_1^{P_1(m)}\cdots T_d^{P_d(m)}x), \qquad x\in X, \end{align} $$

where $Q_{M}:=[M_1]\times \ldots \times [M_k]$ is a rectangle in ${\mathbb {Z}}^k$ . We will often abbreviate $A_{M; X, {\mathcal T}}^{{\mathcal P}}$ to $A_{M; X}^{{\mathcal P}}$ when the tranformations are understood. In some instances, we will write out the averages

$$ \begin{align*}A_{M;X}^{\mathcal P}f(x) \ = \ A_{M_1,.\ldots, M_k;X}^{P_1,\ldots, P_d} f(x) \ \ \ \mathrm{or} \ \ \ A_{{M}; X, {\mathcal T}}^{\mathcal P}f(x) \ = \ A_{M_1,\ldots, M_k;X,T_1,\ldots, T_d}^{P_1,\ldots, P_d} f(x), \end{align*} $$

depending on how explicit we want to be.

Example 1.9. From the point of view of pointwise convergence problems, due to the Calderón transference principle [Reference Calderón19], the most important dynamical system is the integer shift system. Consider the d-dimensional lattice $({\mathbb {Z}}^d, \mathcal B({\mathbb {Z}}^d), \mu _{{\mathbb {Z}}^d})$ equipped with a family of shifts $S_1,\ldots , S_d:{\mathbb {Z}}^d\to {\mathbb {Z}}^d$ , where $\mathcal B({\mathbb {Z}}^d)$ denotes the $\sigma $ -algebra of all subsets of ${\mathbb {Z}}^d$ , $\mu _{{\mathbb {Z}}^d}$ denotes counting measure on ${\mathbb {Z}}^d$ , and $S_j(x)=x-e_j$ for every $x\in {\mathbb {Z}}^d$ (here, $e_j$ is j-th basis vector from the standard basis in ${\mathbb {Z}}^d$ for each $j\in [d]$ ). The average $A_{M; X, {\mathcal T}}^{{\mathcal P}}$ with ${\mathcal T} = (T_1,\ldots , T_d)=(S_1,\ldots , S_d)$ can be rewritten for any $x=(x_1,\ldots , x_d)\in {\mathbb {Z}}^d$ and any finitely supported function $f:{\mathbb {Z}}^d\to \mathbb {C}$ as

(1.10) $$ \begin{align} A_{M; {\mathbb{Z}}^d}^{{\mathcal P}}f(x)=\mathbb{E}_{m\in Q_{M}}f(x_1-P_1(m),\ldots, x_d-P_d(m)). \end{align} $$

The main result of this paper, which implies Conjecture 1.3, is the following ergodic theorem.

For ease of exposition, we only prove Theorem 1.11 in the two-parameter setting $k=2$ , though there are some places in the paper where some arguments are formulated and proved in the multi-parameter setting to convince the reader that our arguments are adaptable to the general multi-parameter setup. However, the patient reader will readily see that all two-parameter arguments are adaptable (at the expense of introducing cumbersome notation, which would make the exposition unreadable) to the general multi-parameter setting for arbitrary $k\ge 2$ , by multiple iterations of the arguments presented in the paper.

We now give some remarks about Theorem 1.11.

1. 1. Theorem 1.11 establishes Conjecture 1.3 for the averages $A_{M; X, T}^{P}f$ . This is the first nontrivial result in the literature establishing pointwise almost everywhere convergence for polynomial ergodic averages in the multi-parameter setting. See [Reference Mirek, Szarek and Wright53] for other pointwise convergence results in the multi-parameter setting.

2. 2. The proof of Theorem 1.11 is relatively simple if $P\in {\mathbb {Z}}[\mathrm {m}_1,\ldots , \mathrm {m}_k]$ is degenerate; see inequality (3.5) in Section 3. We will say that $P\in {\mathbb {Z}}[\mathrm {m}_1,\ldots , \mathrm {m}_k]$ is degenerate if it can be written as

   (1.14) $$ \begin{align} P(\mathrm{m}_1,\ldots, \mathrm{m}_k)=P_1(\mathrm{m}_1)+\ldots+ P_k(\mathrm{m}_k), \end{align} $$
   where $P_1\in {\mathbb {Z}}[\mathrm {m}_1],\ldots , P_k\in {\mathbb {Z}}[\mathrm {m}_k]$ with $P_1(0)=\ldots = P_k(0)=0$ . Otherwise, we say that $P\in {\mathbb {Z}}[\mathrm {m}_1, \ldots , \mathrm {m}_k]$ is non-degenerate. The method of proof of Theorem 1.11 in the degenerate case can be also used to derive quantitative oscillation bounds for the polynomial Dunford and Zygmund theorem establishing (1.7).

3. 3. At the expense of great complexity, one can also prove that inequality (1.13) holds with ${\mathbb {Z}}_+$ in place of $\mathbb {D}_{\tau }$ . However, we do not address this question here, since (1.13) is sufficient for our purposes and will allow us to establish Theorem 1.11(ii).

4. 4. If $(X, \mathcal B(X), \mu )$ is a probability space and the measure preserving transformation T in Theorem 1.11 is totally ergodic, then Theorem 1.11(ii) implies

   (1.15) $$ \begin{align} \lim_{\min\{M_1,\ldots, M_k\}\to\infty}A_{M_1,\ldots, M_k; X, T}^{P}f(x)=\int_Xf(y)d\mu(y) \end{align} $$
   $\mu $ -almost everywhere on X. We recall that a measure preserving transformation T is called ergodic on X if $T^{-1}[B]=B$ implies $\mu (B)=0$ or $\mu (B)=1$ , and totally ergodic if $T^n$ is ergodic for every $n\in {\mathbb {Z}}_+$ .

5. 5. This paper is the first systematic treatment of multi-parameter oscillation semi-norms; see (2.9), Proposition 2.16 and Proposition 2.18. Moreover, it seems that the oscillation semi-norm is the only available tool that allows us to handle efficiently multi-parameter pointwise convergence problems with arithmetic features. This contrasts sharply with the one-parameter setting, where we have a variety of tools including oscillations, variations or jumps to handle pointwise convergence problems; see [Reference Jones, Seeger and Wright38, Reference Mirek, Stein and Trojan49] and the references therein. Multi-parameter oscillations (2.9) were considered for the first time in [Reference Jones, Rosenblatt and Wierdl37] in the context of the Dunford–Zygmund averages (1.1) for commuting measure-preserving transformations.

We close this subsection by emphasizing that the methods developed in this paper allow us to handle averages (1.8) with multiple polynomials. At the expense of some additional work, one can prove the following ergodic theorem.

For simplicity of notation, we have only formulated Theorem 1.16 in the two-parameter setting, but it can be extended to a multi-parameter setting as well. Namely, let $d\ge 2$ and let $(X, \mathcal B(X), \mu )$ be a $\sigma $ -finite measure space equipped with a family of commuting invertible and measure-preserving transformations $T_1,\ldots , T_{d}:X\to X$ . Suppose that $P\in {\mathbb {Z}}[\mathrm {m}_1,\ldots , \mathrm {m}_{d-1}]$ is a polynomial such that

$$ \begin{align*} P(0,\ldots, 0)=\partial_1P(0,\ldots, 0)=\ldots=\partial_{d-1}P(0,\ldots, 0)=0, \end{align*} $$

which has partial degrees (as a polynomial of the variable $\mathrm {m}_i$ for any $i\in [d-1]$ ) at least two. Then the conclusions of Theorem 1.16 remain true for the averages

(1.19) $$ \begin{align} A_{M_1,\ldots, M_{d-1}; X, T_1,\ldots, T_d}^{{\mathrm m}_1,\ldots,{\mathrm m}_{d-1}, P({\mathrm m}_1,\ldots, \mathrm{m}_{d-1})}f \ \ \ \mathrm{in \ place \ of} \ \ \ A_{M_1, M_2; X}^{\mathrm{m}_1,\mathrm{m}_2 , P(\mathrm{m}_1, \mathrm{m}_2)}f. \end{align} $$

All remarks from items 1–4 after Theorem 1.11 remain true for ergodic averages from (1.19). Finally, we emphasize that Theorem 1.11 and Theorem 1.16 make a contribution to the famous Furstenberg–Bergelson–Leibman conjecture, which we now discuss.

1.3 Contributions to the Furstenberg–Bergelson–Leibman conjecture

Furstenberg’s ergodic proof [Reference Furstenberg27] of Szemerédi’s theorem [Reference Szemerédi59] (on the existence arbitrarily long arithmetic progressions in subsets of integers with positive density) was a departure point for modern ergodic Ramsey theory. We refer to the survey articles [Reference Bergelson, Pollicott and Schmidt7], [Reference Bergelson, Hasselblatt and Katok8] and [Reference Frantzikinakis25], where details (including comprehensive historical background) and an extensive literature are given about this fascinating subject. Ergodic Ramsey theory is a very rich body of research, consisting of many natural generalizations of Szemerédi’s theorem, including the celebrated polynomial Szemerédi theorem of Bergelson and Leibman [Reference Bergelson and Leibman9] that motivates the following far-reaching conjecture:

Conjecture 1.20 (Furstenberg–Bergelson–Leibman conjecture [Reference Bergelson and Leibman10, Section 5.5, p. 468])

For given parameters $d, k, n\in \mathbb {N}$ , let $T_1,\ldots , T_d:X\to X$ be a family of invertible measure-preserving transformations of a probability measure space $(X, \mathcal B(X), \mu )$ that generates a nilpotent group of step $l\in {\mathbb {Z}}_{+}$ , and assume that $P_{1, 1},\ldots ,P_{i, j},\ldots , P_{d, n}\in {\mathbb {Z}}[\mathrm m_1,\ldots , \mathrm m_k]$ . Then for any $f_1, \ldots , f_n\in L^{\infty }(X)$ , the nonconventional multiple polynomial averages

(1.21) $$ \begin{align} A_{M; X, T_1,\ldots, T_d}^{P_{1, 1}, \ldots, P_{d, n}}(f_1,\ldots, f_n)(x) =\mathbb{E}_{m\in\prod_{j=1}^k[M_j]}\prod_{j=1}^nf_j(T_1^{P_{1, j}(m)}\cdots T_d^{P_{d, j}(m)} x) \end{align} $$

converge for $\mu $ -almost every $x\in X$ as $\min \{M_1,\ldots , M_k\}\to \infty $ .

Variants of this conjecture were promoted in person by Furstenberg (we refer to Austin’s article [Reference Austin3, pp. 6662]) before it was published by Bergelson and Leibman [Reference Bergelson and Leibman10, Section 5.5, pp. 468] for $k=1$ . The nilpotent and multi-parameter setting is the appropriate setting for Conjecture 1.20 as convergence may fail if the transformations $T_1,\ldots , T_d$ generate a solvable group, as shown by Bergelson and Leibman [Reference Bergelson and Leibman10]. The $L^2(X)$ norm convergence of (1.21) has been studied since Furstenberg’s ergodic proof [Reference Furstenberg27] of Szemerédi’s theorem [Reference Szemerédi59] and is fairly well-understood (even in the setting of nilpotent groups) due to the groundbreaking work of Walsh [Reference Wooley70] with $M_1=\ldots =M_k$ . Prior to Walsh’s paper, extensive efforts had been made towards understanding $L^2(X)$ norm convergence, including breakthrough works of Host–Kra [Reference Host and Kra31], Ziegler [Reference Ziegler71], Bergelson [Reference Bergelson6] and Leibman [Reference Leibman43]. For more details and references, we also refer to [Reference Austin4, Reference Chu, Frantzikinakis and Host21, Reference Frantzikinakis and Kra26, Reference Host and Kra32, Reference Tao61] and the survey articles [Reference Bergelson, Pollicott and Schmidt7, Reference Bergelson, Hasselblatt and Katok8, Reference Frantzikinakis25].

The situation is dramatically different for the pointwise convergence problem (1.21), but recently, significant progress has been made towards establishing the Furstenberg–Bergelson–Leibman conjecture. Now let us make a few remarks about this conjecture, its history and the current state of the art.

1. 1. The case $d=k=n=1$ of Conjecture 1.20 with $P_{1, 1}(m)=m$ follows from Birkhoff’s ergodic theorem [Reference Birkhoff11]. In fact, the almost everywhere limit (as well as the norm limit; see also [Reference von Neumann64]) of (1.21) exists also for all functions $f\in L^p(X)$ , with $1\le p<\infty $ , defined on any $\sigma $ -finite measure space $(X, \mathcal B(X), \mu )$ .

2. 2. The case $d=k=n=1$ of Conjecture 1.20 with arbitrary polynomials $P_{1, 1}\in {\mathbb {Z}}[\mathrm {m}]$ (as we have seen above) was the famous open problem of Bellow [Reference Bellow5] and Furstenberg [Reference Furstenberg24], which was solved by the first author [Reference Bourgain13, Reference Bourgain14, Reference Bourgain15] in the mid 1980s. In fact, in [Reference Bourgain13, Reference Bourgain14, Reference Bourgain15], it was shown that the almost everywhere limit (as well as the norm limit; see also [Reference Furstenberg29]) of (1.21) exists also for all functions $f\in L^p(X)$ , with $1< p<\infty $ , defined on any $\sigma $ -finite measure space $(X, \mathcal B(X), \mu )$ . In contrast to the Birkhoff theorem, if $P_{1,1}\in {\mathbb {Z}}[\mathrm {n}]$ is a polynomial of degree at least two, the pointwise convergence at the endpoint for $p=1$ may fail as was shown by Buczolich and Mauldin [Reference Buczolich and Mauldin18] for $P_{1,1}(m)=m^2$ and by LaVictoire [Reference LaVictoire42] for $P_{1,1}(m)=m^k$ for any $k\ge 2$ .

3. 3. In the commutative case (step $\ell = 1$ ) where $d,k\in {\mathbb {Z}}_+$ and $n=1$ of Conjecture 1.20 with arbitrary polynomials $P_{1,1},\ldots , P_{d,1}\in {\mathbb {Z}}[\mathrm {m}_1, \ldots , \mathrm {m}_k]$ in the diagonal setting $M_1=\ldots =M_k$ – that is, the multi-dimensional one-parameter setting – was solved by the second author with Trojan in [Reference Mirek and Trojan54]. As before, it was shown that the almost everywhere limit (as well as the norm limit) of (1.21) exists also for all functions $f\in L^p(X)$ , with $1< p<\infty $ , defined on any $\sigma $ -finite measure space $(X, \mathcal B(X), \mu )$ .

4. 4. The question to what extent one can relax the commutation relations between $T_1,\ldots , T_d$ in (1.21), even in the one-parameter case $M_1=\ldots =M_k$ , is very intriguing. Some particular examples of averages (1.21) with $d,k\in {\mathbb {Z}}_+$ and $n=1$ and polynomial mappings with degree at most two in the step two nilpotent setting were studied in [Reference Ionescu, Magyar, Stein and Wainger33, Reference Magyar, Stein and Wainger45]. Recently, the second author with Ionescu, Magyar and Szarek [Reference Ionescu, Magyar, Mirek and Szarek36] established Conjecture 1.20 with $d\in {\mathbb {Z}}_+$ and $k=n=1$ and arbitrary polynomials $P_{1,1},\ldots , P_{d,1}\in {\mathbb {Z}}[\mathrm {m}]$ in the nilpotent setting (i.e., when $T_1,\ldots , T_{d}:X\to X$ is a family of invertible measure-preserving transformations of a $\sigma $ -finite measure space $(X, \mathcal B(X), \mu )$ that generates a nilpotent group of step two).

5. 5. In contrast to the commutative linear theory, the multilinear theory is wide open. Only a few results are known in the bilinear $n=2$ and commutative $d=k=1$ setting. The first author [Reference Bourgain16] established pointwise convergence when $P_{1,1}(m)=am$ and $P_{1,2}(m)=bm$ , with $a, b\in {\mathbb {Z}}$ . Recently, the third author with Krause and Tao [Reference Krause, Mirek and Tao41] proved pointwise convergence for the polynomial Furstenberg–Weiss averages [Reference Furstenberg28, Reference Furstenberg and Weiss30] corresponding to $P_{1,1}(m)=m$ and $P_{1, 2}(m)=P(m)$ with $P\in {\mathbb {Z}}[\mathrm {m}]$ and $\mathrm {deg }\,P\ge 2$ .

6. 6. A genuinely multi-parameter case $d=k\ge 2$ with $n=1$ of Conjecture 1.20 for averages (1.21) with linear orbits (i.e. $P_{j,1}(m_1, \ldots , m_d)=m_j$ for $j\in [d]$ ) was established independently by Dunford [Reference Dunford23] and Zygmund [Reference Zygmund72] in the early 1950s. Moreover, it follows from [Reference Dunford23, Reference Zygmund72] that the almost everywhere convergence (as well as the norm convergence) of (1.21) holds for all functions $f\in L^p(X)$ , with $1< p<\infty $ , defined on any $\sigma $ -finite measure space $(X, \mathcal B(X), \mu )$ equipped with a family of measure-preserving transformations $T_1,\ldots , T_d:X\to X$ , which does not need to be commutative. One also knows that pointwise convergence fails if $p=1$ . A polynomial variant of the Dunford and Zygmund theorem was discussed above; see (1.7).

We close this discussion by emphasizing that Theorem 1.11 and Theorem 1.16 also contribute to the Furstenberg–Bergelson–Leibman conjecture and, together with all the results listed above, support the evidence that Conjecture 1.20 may be true in full generality though a complete solution seems very difficult.

1.4 Overview of the paper

The paper is organized as follows. In Section 2, we fix necessary notation and terminology. We also introduce the definition of multi-parameter oscillations (2.9) and collect their useful properties; see Proposition 2.16 and Proposition 2.18. In Section 3, we give a detailed proof of Theorem 1.11 by reducing the matter to oscillation estimates for truncated variants of averages $A_{M_1, M_2; X}^{ P}f$ ; see definition (3.6) and Theorem 3.16, which in turn is reduced to the integer shift system, and Theorem 3.19. A result that may be of independent interest is Proposition 3.7, which shows that oscillations for $A_{M_1, M_2; X}^{P}f$ and their truncated variants are, in fact, comparable. In Section 3, see inequality (3.5), and we also illustrate how to prove Theorem 1.11 in the degenerate case in the sense of definition (1.14) stated after Theorem 1.11. These arguments can be also used to prove oscillation bounds for the polynomial Dunford and Zygmund theorem, which in turn imply (1.7).

We start with a brief overview of the proof of Theorem 3.19, which implies Theorem 1.5 when $k=2$ and takes up the bulk of this paper. The proof requires substantial new ideas to overcome a series of new difficulties arising in the multi-parameter setting. These complications do not arise in the one-parameter setup [Reference Bourgain13, Reference Bourgain14, Reference Bourgain15]. The most notable obstacle is the lack of nestedness in the definition of averaging operators (1.8) when the parameters $M_1,\ldots , M_k$ are allowed to run independently. The lack of nestedness complicates every argument in the circle method, which is the main tool in these kinds of problems. In order to understand how the lack of nestedness may affect the underlying arguments, it will be convenient to illustrate this phenomenon by comparing Figure 1 and Figure 2 below. The first picture (Figure 1) represents the family of nested cubes, which is increasing when the time parameter increases. The diagonal relation between parameters $M_1=\ldots =M_k$ is critical.

The second picture (Figure 2) represents the family which is genuinely multi-parameter and there is no nestedness as the parameters $M_1,\ldots , M_k$ vary independently.

Our remedy to overcome the lack of nestedness will be to develop the so-called multi-parameter circle method, which will be based on an iterative implementation of the classical circle method. Although this idea sounds very simple, it is fairly challenging to formalize it in the context of Conjecture 1.3. We remark that the multi-parameter circle method has been developed for many years in the context of various problems arising in number theory (see [Reference Arkhipov, Chubarikov and Karatsuba1] for more details and references, including a comprehensive historical background) though it is not applicable directly in the ergodic context. We now highlight the key ingredients that we develop in this paper and that will lead us to develop the multi-parameter circle method in the context of Theorem 3.19:

1. (i) ‘Backwards’ Newton diagram is the key tool allowing us to overcome the problem with the lack of nestedness. In particular, it permits us to understand geometric properties of the underlying polynomials in Theorem 3.19 by extracting dominating monomials. The latter are critical in making a distinction between minor and major arcs in the multi-parameter circle method. As far as we know, this is the first time when the concept of Newton diagrams is exploited in problems concerning pointwise ergodic theory. We refer to Section 4 for details.

2. (ii) We derive new estimates for multi-parameter exponential sums arising in the analysis of Fourier multipliers corresponding to averages (1.10). In Section 5, we build a theory of double exponential sums, which is dictated by the geometry of the corresponding ‘backwards’ Newton diagrams. Although the theory of multi-parameter exponential sums is rich (see for example, [Reference Arkhipov, Chubarikov and Karatsuba1]), our results seem to be new and the idea of exploiting ‘backwards’ Newton diagrams and iterative applications of the Vinogradov mean value theorem [Reference Bourgain, Demeter and Guth17] in estimates of exponential sums is quite efficient.

3. (iii) A multi-parameter Ionescu–Wainger multiplier theory is developed in Section 6. The Ionescu–Wainger multiplier theorem [Reference Ionescu and Wainger34] was originally proved for linear operators; see also [Reference Mirek46, Reference Mirek, Stein and Zorin-Kranich52, Reference Pierce55, Reference Tao62]. In this paper, we prove a semi-norm variant of the Ionescu–Wainger theory in the one-parameter setting, which is consequently upgraded to the multi-parameter setup. ‘Backwards’ Newton diagrams play an essential role in our considerations here as well.

4. (iv) Finally, we arrive at the stage where the multi-parameter circle method is feasible by a delicate iterative application of the classical circle method. In this part of the argument, the lack of nestedness is particularly unpleasant, causing serious difficulties in controlling error terms that arise in estimating contributions of the corresponding Fourier multipliers on minor and major arcs, which are genuinely multi-parameter. In Section 7, we illustrate how one can use all the tools developed in the previous sections to give a rigorous proof of Theorem 3.19.

We now take a closer look at the tools highlighted above. In Section 4, we introduce the concept of ‘backwards’ Newton diagram, which is the key to circumvent the difficulties caused by the lack of nestedness. The ‘backwards’ Newton diagram splits the parameter space into a finite number of sectors, where certain relations between parameters are given. In each of these sectors there is a dominating monomial which, in turn, gives rise to an implementation of the circle method to each of the sectors separately. The distinctions between minor and major arcs are then dictated by the degree of the associated dominating monomial. At this stage, we eliminate minor arcs by invoking estimates of double exponential sums from Proposition 5.37. This proposition is essential in our argument; its proof is given in Section 5. The key ingredients are Proposition 5.22, which may be thought of as a two parameter counterpart of the classical Weyl’s inequality, and the properties of the ‘backwards’ Newton diagram. Although the theory of multi-parameter exponential sums has been developed over the years (see [Reference Arkhipov, Chubarikov and Karatsuba1] for a comprehensive treatment of the subject), we require more delicate estimates than those available in the existing literature. In this paper, we give an ad-hoc proof of Proposition 5.22, which follows from an iterative application of Vinogradov’s mean value theorem, and may be interesting in its own right. In Section 5, we also develop estimates for complete exponential sums. In Section 6, we develop the Ionescu–Wainger multiplier theory for various semi-norms in one-parameter as well as in multi-parameter settings. Our result in the one-parameter setting, Theorem 6.14, is formulated for oscillations and maximal functions, but the proofs also work for $\rho $ -variations or jumps. In fact, Theorem 6.14 is the starting point for establishing the corresponding multi-parameter Ionescu–Wainger theory for oscillations. The latter theorem will be directly applicable in the analysis of multipliers associated with the averages $A_{M_1, M_2; X}^{P}f$ . The results of Section 6 are critical in our multi-parameter circle method that is presented in Section 7, as it allows us to efficiently control the error terms that arise on major arcs as well as the contribution coming from the main part. In contrast to the one-parameter theory [Reference Bourgain13, Reference Bourgain14, Reference Bourgain15], the challenge here is to control, for instance, maximal functions corresponding to error terms. For this purpose, all error terms have to be provided with asymptotic precision, which usually requires careful arguments. The details of the multi-parameter circle method are presented in Section 7 in the context of the proof of Theorem 3.19.

1.5 More about Conjecture 1.20

Conjecture 1.20 is one of the major open problems in pointwise ergodic theory, which seems to be very difficult due to its multilinear nature. Here, in light of the Arkhipov, Chubarikov and Karatsuba [Reference Arkhipov, Chubarikov and Karatsuba2] equidistribution theory which works also for multiple polynomials, it seems reasonable to propose a slightly more modest problem (implied by Conjecture 1.20) though still very interesting and challenging that can be subsumed under the following conjecture:

Conjecture 1.22. Let $d, k\in {\mathbb {Z}}_+$ be given and let $(X, \mathcal B(X), \mu )$ be a probability measure space endowed with a family of invertible commuting measure-preserving transformations $T_1,\ldots , T_d:X\to X$ . Assume that $P_1,\ldots , P_d \in {\mathbb {Z}}[\mathrm {m}_1, \ldots , \mathrm {m}_k]$ . Then for any $f\in L^{\infty }(X)$ , the multi-parameter linear polynomial averages

$$ \begin{align*} A_{M_1,\ldots, M_k; X, T_1,\ldots, T_d}^{P_1, \ldots, P_d}f(x)=\mathbb{E}_{m\in \prod_{j=1}^k[M_j]}f(T_1^{P_1(m)}\cdots T_d^{P_d(m)}x) \end{align*} $$

converge for $\mu $ -almost every $x\in X$ , as $\min \{M_1,\ldots , M_k\}\to \infty $ .

Even though we prove Conjecture 1.3 here, it is not clear whether Conjecture 1.22 is true for all polynomials. If it is not true for all polynomials, it would be interesting, in view of Theorem 1.16, to characterize the class of those polynomials for which Conjecture 1.22 holds. Although the averages from Theorem 1.11 and Theorem 1.16 share a lot of difficulties that arise in the general case, there are some cases that are not covered by the methods of this paper. An interesting difficulty arises for the so-called partially complete exponential sums when we are seeking estimates of the form

(1.23) $$ \begin{align} \frac{1}{M_1q}\sum_{m_1=1}^{M_1}\Big|\sum_{m_2=1}^q\boldsymbol{e}(a_2m_2/q+a_3P(m_1, m_2)/q)\Big|\lesssim q^{-\delta}, \end{align} $$

for all $M_1, q\in {\mathbb {Z}}_+$ and some $\delta \in (0,1)$ , whenever $(a_2, a_3, q)=1$ . These kinds of estimates arise from applications of the circle method with respect to the second variable $m_2$ for the averages $A_{M_1, M_2; X}^{\mathrm {m}_1,\mathrm {m}_2 , P(\mathrm {m}_1, \mathrm {m}_2)}f$ when we are at the stage of applying the circle method with respect to the first variable $m_1$ . Here, the assumption that P has partial degrees (as a polynomial of the variable $m_1$ and a polynomial of the variable $m_2$ ) at least two is essential. Otherwise, if $M_1<q$ , the decay $q^{-\delta }$ in (1.23) is not possible. In order to see this, it suffices to take $P(m_1, m_2)=m_1^2m_2$ . A proof of Theorem 1.16 for polynomials of this type, as well as Conjecture 1.22, will require a deeper understanding and substantially new methods. We believe that the proof of Theorem 1.11 is an important contribution towards understanding Conjecture 1.22 that may shed new light on the general case and either lead to its full resolution or to a counterexample. The second and fourth authors plan to pursue this problem in the future.

1.6 In Memoriam

It was a great privilege and an unforgettable experience for the second and fourth authors to know and work with Elias M. Stein (January 13, 1931–December 23, 2018) and Jean Bourgain (February 28, 1954–December 22, 2018). Eli and Jean had an immeasurable effect on our lives and careers. It was a very sad time for us when we learned that Eli and Jean passed away within an interval of one day in December 2018. We miss our friends and collaborators dearly.

We now briefly describe how the collaboration on this project arose. In 2011, the second and fourth authors started to work on some aspects of a multi-parameter circle method in the context of various discrete multi-parameter operators. These efforts resulted in a draft on estimates for certain two-parameter exponential sums. This draft was sent to the first author sometime in the first part of 2016. In October 2016, when the second author was a member of the Institute for Advanced Study, it was realized (during a discussion between the first two authors) that the estimates from this draft are closely related to a multi-parameter Vinogradov’s mean value theorem. This was interesting to the first author who at that time was involved in developing the theory of decoupling. We also realized that some ideas of a multi-parameter circle method from the draft of the second and fourth authors may be upgraded and used in attacking a multi-parameter variant of the Bellow and Furstenberg problem formulated in Conjecture 1.3. That was the first time when the second, third and fourth authors learned about this conjecture and unpublished observations of the first author from the late 1980s that resulted in establishing pointwise convergence in (1.7). This was the starting point of our collaboration. At that time another question arose, which is also related to this paper. It is interesting whether a sharp multi-parameter variant of Vinogradov’s mean value theorem can be proved using the recent developments in the decoupling theory from [Reference Bourgain, Demeter and Guth17]. A multi-parameter Vinogradov’s mean value theorem was investigated in [Reference Arkhipov, Chubarikov and Karatsuba1], but the bounds are not optimal. So the question is about adapting the methods from [Reference Bourgain, Demeter and Guth17] to the multi-parameter setting in order to obtain sharp bounds, and their applications in the exponential sum estimates.

A substantial part of this project was completed at the end of November/beginning of December 2016, when the fourth author visited Princeton University and the Institute for Advanced Study. At that time, we discussed (more or less) all tools that were needed to establish Theorem 1.11 for the monomial $P(m_1, m_2)=m_1^2m_2^3$ . Then we were convinced that we could establish Conjecture 1.22 in full generality, but various difficulties arose when we started to work out the details, and we ultimately only managed to prove Theorem 1.11 and Theorem 1.16. The second and fourth authors decided to illustrate the arguments in the two-parameter setting and the reason is twofold. On the one hand, we wanted to avoid introducing heavy multi-parameter notation capturing all combinatorial nuances arising in this project. On the other hand, what is more important is that we wanted to illustrate the spirit of our discussions that took place in 2016. For instance, the arguments presented in Section 5 can be derived by using Weyl differencing argument, which may be even simpler and can be easily adapted to the multi-parameter setting, though our presentation is very close to the arguments that we developed in 2016, and also motivates the question about the role of decoupling theory in the multi-parameter Vinogradov’s mean value theorem that we have stated above.