Pointwise ergodic theorems for arithmetic sets

Jean Bourgain

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Ergodic theory and connections with analysis and probability.

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Counting and convergence in ergodic theory

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Appendix on return-time sequences

Jean Bourgain, Harry Furstenberg, Yitzhak Katznelson, Donald S. Ornstein (1989)

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On the ergodic theorems (I) (Generalized ergodic theorems)

C. Ryll-Nardzewski (1951)

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Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations

Ryotaro Sato (1995)

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Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average $n^{- 1} \sum_{i = 0}^{n - 1} f \circ τ^{i} (x)$ converges almost everywhere to a function f* in $L (p_{1}, q_{1}]$ , where (pq) and $(p_{1}, q_{1}]$ are assumed to be in the set $(r, s) : r = s = 1, o r 1 < r < \infty a n d 1 \leq s \leq \infty, o r r = s = \infty$ . Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized...