*(English)*Zbl 0910.28013

Let $(X,\mathcal{F},\mu ,T)$ be a dynamical systems consisting of a measure-preserving transformation defined on a standard probability space. A bounded sequence $\left\{{a}_{i}\right\}$ of complex numbers are called universal weights (for the pointwise ergodic theorem) if for any dynamical system $(Y,\mathcal{G},\nu ,S)$ and $g\in {L}^{1}\left(\nu \right)$, $\frac{1}{N}{\sum}_{j=0}^{N-1}{a}_{i}g\left({S}^{j}y\right)$ converges.

The following theorem was proved by *J. Bourgain*, *H. Furstenberg*, *Y. Katznelson* and *D. S. Ornstein* [Publ. Math., Inst. Haut. Étud. Sci. 69, 5-45 (1989; Zbl 0705.28008)] (and a joinings proof was given by *D. J. Rudolph* [Ergodic Theory Dyn. Syst. 14, No. 1, 197-203 (1994; Zbl 0799.28010)]):

For any dynamical system $(X,\mathcal{F},\mu ,T)$ and $f\in {L}^{\infty}\left(\mu \right)$ for $\mu $ a.e. $x$, the sequence of values $f\left({T}^{i}x\right)$ are universal weights.

The author proves a multi-term version of the above result first proposed by I. Assani, i.e., involving averages of the form

where all the ${f}_{i}$ are bounded and where for each $j<k$, the points ${x}_{1},{x}_{2},\cdots ,{x}_{j}$ guaranteeing convergence can be chosen universally, without knowledge of the transformations and functions to follow. The author is able to avoid the ${L}^{2}$ orthogonality arguments in the proof of Bourgain et al., which requires the splitting of the ${L}^{2}$ space into Kronecker functions and their orthocomplement, by using the notion of pointwise genericity. In particular, a disjointness result on joinings avoids the need to identify distinguished factor algebras for the higher term averages. A disadvantage of this approach is that delicate information about characteristic factors is not available (as in, for example, the work of *D. J. Rudolph* [Lond. Math. Soc. Lect. Note Ser. 205, 369-432 (1995; Zbl 0877.28012)], concerning the Conze-Lesigne algebra).

##### MSC:

28D05 | Measure-preserving transformations |