Let be a dynamical systems consisting of a measure-preserving transformation defined on a standard probability space. A bounded sequence of complex numbers are called universal weights (for the pointwise ergodic theorem) if for any dynamical system and , 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 and for a.e. , the sequence of values 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 are bounded and where for each , the points guaranteeing convergence can be chosen universally, without knowledge of the transformations and functions to follow. The author is able to avoid the orthogonality arguments in the proof of Bourgain et al., which requires the splitting of the 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).