Pointwise almost-everywhere convergence of multiple ergodic averages for commuting transformations

For commuting invertible measure-preserving transformations $T_1,\dots,T_\ell$ of a probability space, Tao (2008) proved that the multiple ergodic averages $(1/N)\sum_{n<N} f_1(T_1^n x)\cdots f_\ell(T_\ell^n x)$ converge in $L^2$, and Walsh (2012) extended norm convergence to nilpotent groups; the pointwise counterpart remains unknown. Pointwise almost-everywhere convergence of these averages is still open already for $\ell=2$ commuting transformations; it is the linear case of Bergelson's 1996 Question 9 and of the pointwise Furstenberg--Bergelson--Leibman conjecture. Partial results exist only for distal systems, for distinct-degree polynomial iterates, and for single-transformation bilinear averages.

Background

A measure-preserving transformation of a probability space $(X,\mathcal{X},\mu)$ is an invertible map $T:X\to X$ with $\mu(T^{-1}A)=\mu(A)$ for all measurable $A$. The classical ergodic theorems describe the behavior of time averages $\frac{1}{N}\sum_{n=0}^{N-1} f(T^n x)$: von Neumann's mean ergodic theorem gives convergence in $L^2(\mu)$, while Birkhoff's pointwise ergodic theorem gives convergence for $\mu$-almost every $x$. Both conclusions become far harder, and genuinely different, when several transformations are averaged together.

For commuting measure-preserving transformations $T_1,\dots,T_\ell$ and bounded measurable $f_1,\dots,f_\ell\in L^\infty(\mu)$, the multiple (also called non-conventional or Furstenberg) ergodic averages are

$$A_N(x)=\frac{1}{N}\sum_{n=0}^{N-1} f_1(T_1^n x)\cdots f_\ell(T_\ell^n x).$$

These objects are central to ergodic Ramsey theory: through the Furstenberg correspondence principle, their limits control multiple recurrence, which in turn encodes arithmetic patterns such as arithmetic progressions in positive-density sets. The $L^2$ (norm) convergence of $A_N$ for arbitrary commuting $T_i$ was proved by Tao (2008) by a finitary, hypergraph-regularity-style argument (a discrete analogue of the decomposition of uniform hypergraphs into structured and pseudorandom parts), and Walsh (2012) generalized norm convergence to transformations generating a nilpotent group (commuting transformations generate an abelian, hence nilpotent, group, and form the basic case). So the norm question is completely settled, and the surviving issue is pointwise behavior.

Pointwise convergence is classically approached through maximal inequalities: a bound of the form $\|\sup_{N\ge 1}|A_N|\|_{L^p(\mu)}\le C\,\|f_1\|_{L^{q_1}}\cdots\|f_\ell\|_{L^{q_\ell}}$ with finite exponents implies almost-everywhere convergence by standard density arguments, in direct analogy with the hard analysis behind Birkhoff's theorem. In the single-transformation case $T_i=T^i$, Bourgain (1990) established the required bilinear maximal inequality for $\ell=2$, proving almost-everywhere convergence of $\frac{1}{N}\sum_{n<N} f(T^n x)g(T^{2n}x)$; the polynomial bilinear extension was completed only recently by Krause, Mirek and Tao (2022) using the Ionescu--Wainger multiplier theorem (a harmonic-analysis result supplying $L^p$ bounds for multiplier operators adapted to sets of canonical fractions of integers). For genuinely distinct commuting transformations, no such multilinear maximal inequality is known, and even the case of two commuting maps is a recognized open problem. Bergelson asked in 1996 (Question 9 of his survey 'Ergodic Ramsey theory -- an update') whether the polynomial-iterate version of these averages converges pointwise for commuting transformations, and the pointwise form of the Furstenberg--Bergelson--Leibman conjecture (transformations generating a nilpotent group, arbitrary integer-polynomial iterates) is the standard modern formulation of this question.

Known partial results delineate the frontier: pointwise convergence holds for distal systems (systems built as towers of compact extensions, a structured class introduced in Furstenberg's work) by work of Huang--Shao--Ye and Donoso--Sun; it holds for commuting transformations along polynomial iterates of pairwise distinct degrees by the multilinear circle method of Kosz--Mirek--Peluse--Wan--Wright (2024) (a circle-method-style decomposition of frequency space adapted to multilinear averages); and it holds in the bilinear single-transformation polynomial regime. None of these covers the original linear averages for general commuting systems, because all linear iterates share the same degree. The requirement that the transformations commute is essential: Austin (2025) constructed non-commuting pairs for which the double averages fail to converge even in $L^2$.

Problem Statement

Let $\ell\ge 2$ and let $T_1,\dots,T_\ell$ be commuting invertible measure-preserving transformations of a probability space $(X,\mathcal{X},\mu)$. For functions $f_1,\dots,f_\ell\in L^\infty(\mu)$, consider the multiple ergodic averages

$$A_N(x)=\frac{1}{N}\sum_{n=0}^{N-1} f_1(T_1^n x)\, f_2(T_2^n x)\cdots f_\ell(T_\ell^n x),\qquad N=1,2,\dots$$

Question: do the averages $A_N(x)$ converge as $N\to\infty$ for $\mu$-almost every $x\in X$, for every $\ell\ge 2$, every commuting family $T_1,\dots,T_\ell$, and all $f_1,\dots,f_\ell\in L^\infty(\mu)$? This is the pointwise counterpart of Tao's 2008 norm-convergence theorem (the unnumbered closing remark of Section 1 of his paper) and the linear-iterate case of Bergelson's 1996 Question 9 and of the pointwise Furstenberg–Bergelson–Leibman conjecture. It is open already for $\ell=2$, i.e. for two commuting transformations $S,T$ and the averages $\frac{1}{N}\sum_{n=0}^{N-1} f(S^n x)\,g(T^n x)$. The task's target is this $\ell=2$ instance, the minimal open case: do the double averages $\frac{1}{N}\sum_{n=0}^{N-1} f(S^n x)\,g(T^n x)$ converge $\mu$-almost everywhere for all commuting invertible measure-preserving $S,T$ and all $f,g\in L^\infty(\mu)$? A negative answer requires an explicit commuting system and bounded functions for which $A_N(x)$ fails to converge on a set of positive $\mu$-measure. No restrictions on $\ell$, ergodicity, entropy, or the class of commuting systems are admitted in this general problem, of which the targeted $\ell=2$ instance is a sub-case.

References

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