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Fully generic sequences and a multiple-term return-times theorem. (English) Zbl 0910.28013

Let $\left(X,ℱ,\mu ,T\right)$ 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 $\left(Y,𝒢,\nu ,S\right)$ 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 $\left(X,ℱ,\mu ,T\right)$ 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

$\frac{1}{N}\sum _{i=0}^{N-1}{f}_{1}\left({T}_{1}^{i}{x}_{1}\right){f}_{2}\left({T}_{2}^{i}{x}_{2}\right)\cdots {f}_{k}\left({T}_{k}^{i}{x}_{k}\right),$

where all the ${f}_{i}$ are bounded and where for each $j, 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