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Ergodic averages and the large intersection property along IP sets
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abstract
We study multiple ergodic averages along IP sets, meaning we restrict iterates in the averages to all finite sums of some infinite sequence of natural numbers. We give criteria for convergence and divergence in mean of these multiple averages and derive sufficient conditions for convergence to the projection onto the space of invariant functions. For a class of sequences that, roughly speaking, only have rational obstructions to such a limit, we show that the behavior is controlled by nilsystems. We also consider pointwise convergence, obtaining convergence and a formula for a set of functions on nilsystems that are dense in $L^2$. Finally, we show that certain correlations have optimally large intersections along an IP set
Forward citations
Cited by 2 Pith papers
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Pointwise Convergence of Ergodic Averages Along Integer Cantor Sets
Ergodic averages along integer Cantor sets converge almost everywhere for L^p functions, p≥2.
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$\mathrm{IP}_{\mathrm{rat}}$-polynomial recurrence and large intersections
For rationally independent nonlinear polynomials, the set of n giving nearly independent intersections of a positive-density set has positive lower IP_rat density.
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