The 2024 preprint of Kosz, Mirek, Peluse, Wan and Wright, now accepted by the Annals of Mathematics, gives a major new answer to a question of Bergelson on pointwise convergence of multilinear polynomial ergodic averages (Kosz et al. 2024). In the distinct-degree case, they prove pointwise almost everywhere convergence for an arbitrary number of commuting transformations, and in fact obtain a stronger quantitative theorem: convergence holds for functions in the natural Hölder range .
Let be a probability space, so , and let be a measure-preserving transformation, meaning that is measurable and If is invertible, then is defined for every , with , for , and for . For , denotes the space of measurable functions with and denotes the space of essentially bounded functions. A sequence of functions converges in if there is such that . It converges pointwise almost everywhere if, outside a set of measure zero, the numerical sequence converges.
Consider the multilinear polynomial ergodic averages where are polynomials with integer coefficients, are invertible measure-preserving transformations, and are measurable functions on . The assumption guarantees that for every , so the iterates are well defined. The transformations are said to commute if , or equivalently for every .
The average is called “multilinear” because it depends linearly on each function separately, and it is called “polynomial” because the orbit of is sampled at the polynomial times . When and , this is the classical Birkhoff average. For , it is a non-conventional average: instead of following one observable along one orbit, it compares several observables along several polynomially parametrized orbits at the same time. These averages are central in ergodic Ramsey theory. For example, if , then integration of involves quantities of the form and, after adjoining the extra factor , such expressions measure polynomial multiple recurrence of a set . In finite cyclic models they count polynomial configurations in dense subsets of integers. Thus convergence of says that these recurrence statistics stabilize as .
In 1996, Bergelson (Bergelson 1996) asked whether converges, first in the norm and then pointwise almost everywhere, in the case when all are bounded and all are pairwise commuting. This fundamental question shaped much of the modern research on multiple ergodic averages.
We first discuss norm convergence. In 2005, Leibman (Leibman 2005) proved the -convergence of in the single-transformation case . In 2008, Tao (Tao 2008) resolved the commuting-transformation case when all the polynomials are linear. In 2011, Chu, Frantzikinakis and Host (Chu et al. 2011) resolved the case when the polynomials have distinct degrees. Finally, in 2012 Walsh (Walsh 2012) answered Bergelson’s norm-convergence question in full. In fact, Walsh proved -convergence for a substantially more general class of averages, in which the transformations need not commute but are required to generate a nilpotent group.
Invertible measure-preserving transformations form a group under composition. In an abstract group , the commutator of two elements is The lower central series of is the decreasing sequence of subgroups where is generated by all commutators with and . The group is called nilpotent if this sequence eventually reaches the trivial subgroup , where is the identity element. Equivalently, there is some such that . Abelian groups are exactly the nilpotent groups of step at most one.
The average is the special case of Theorem obtained by taking , , and whenever . Bergelson’s original commuting-transformation setting corresponds to the special case in which the group generated by is abelian. As noted in (Bergelson and Leibman 2002), the nilpotency assumption in Walsh’s theorem cannot simply be removed: convergence can fail even for , bounded functions, and the linear choice .
Walsh’s theorem proves that the -limit exists, but in this generality it does not identify the limit explicitly. Recently, Frantzikinakis and Kuca (Frantzikinakis and Kuca 2025) studied the limit of for commuting transformations and bounded functions. Under additional hypotheses on the polynomial family, they identify characteristic factors and obtain concrete limit formulas. For instance, in the linearly independent case, the rational Kronecker factor is characteristic; in particular, for totally ergodic transformations the limit is the product of the integrals of the functions.
Pointwise convergence is much more delicate. Norm convergence is a global statement in , while pointwise convergence asks for a limiting law along almost every individual orbit. The case and is Birkhoff’s ergodic theorem (Birkhoff 1931). The case and arbitrary is Bourgain’s polynomial ergodic theorem (Bourgain 1989). In 1990, Bourgain (Bourgain 1990) also resolved the bilinear single-transformation linear case , , , and . Equivalently, he established pointwise almost everywhere convergence of when are bounded and are linear polynomials. In 2022, Krause, Mirek and Tao (Krause et al. 2022) treated a much harder bilinear case in which one polynomial is linear and the other is nonlinear.
Theorem is very general: it holds even on -finite measure spaces, not just probability spaces. Nevertheless, it was new even for probability spaces and bounded functions. The special case answers Problem 11 from Frantzikinakis’ open problems survey (Frantzikinakis 2016). The endpoint cases are genuinely different: the counterexamples discussed in Theorem show that one cannot expect Theorem to extend in general to or .
On a probability space, the case of Theorem reduces to Bourgain’s theorem (Bourgain 1989) on pointwise convergence along polynomial iterates. The proof of Theorem combines Bourgain’s harmonic-analytic ideas with inverse theorems from additive combinatorics and, at large scales, harmonic analysis on the adelic integers.
Kosz, Mirek, Peluse, Wan and Wright (Kosz et al. 2024) have now proved a far-reaching multilinear extension in the distinct-degree case. Their theorem applies to any number of functions and transformations, and it goes beyond the bounded setting by allowing functions in the natural Hölder range.
The bounded case in Bergelson’s question follows immediately from Theorem : on a probability space, every bounded function belongs to for every finite , and one may choose large enough so that . Thus Bergelson’s pointwise convergence problem is now solved for arbitrary multilinearity in the important case when the polynomial iterates have distinct degrees.
This leaves two natural frontiers. The first is to remove the distinct-degree assumption, since repeated degrees are precisely where new arithmetic resonances appear. The second is to move from commuting transformations toward the nilpotent setting suggested by Walsh’s norm-convergence theorem and by the Furstenberg–Bergelson–Leibman conjecture. The result of Kosz, Mirek, Peluse, Wan and Wright does not close these problems, but it changes the landscape: for the first time, pointwise convergence of general multilinear polynomial averages is known beyond the previously accessible bilinear and single-transformation regimes.