Polynomial ergodic averages for certain countable ring actions

Abstract

A recent result of Frantzikinakis in [Fra3] establishes sufficient conditions for joint ergodicity in the setting of Z-actions. We generalize this result for actions of secondcountable locally compact abelian groups. As an application, we show that, given an ergodic action (T n ) n∈F of a countable field F with characteristic zero on a probability space (X, B, µ) and a family {p 1 , . . . , p k } of F -linearly independent essentially distinct polynomials, we havewhere f i ∈ L ∞ (µ), (Φ N ) is a Følner sequence of (F, +), and the convergence takes place in L 2 (µ). We also obtain corollaries in combinatorics and topological dynamics.

Joint ergodicity of fractional powers of primes

“…where a ∈ R + is not an integer. We do expect the L 2 (µ)-limit of the averages (5) to be equal to the L 2 (µ)-limit of the averages 1…”

“…Lastly, we remark that although the reduction offered by Theorem 2.1 is very helpful when dealing with averages with independent iterates, as is the case in (3), it does not offer any help when the iterates are linearly dependent, which is the case for the averages (5) 1…”

“…A crucial ingredient used in the proof of our main result is the following result that gives convenient necessary and sufficient conditions for joint ergodicity of a collection of sequences (see also [5] for an extension of this result for sequences b 1 , . .…”

“…Note that for fixed n, h ∈ N, when n 1 ranges in J n,h the value of p 1 (h)n 0.5 1 ranges in an interval of length at most 1 and the same property holds for the values of p 2 (h)n 0. 5 1 , q 1 (h)n 0.1 1 , q 2 (h)n 0.1 1 . Hence, for n 1 ∈ J n,h we have…”

“…Although the method of Theorem 2.4 does give good seminorm bounds for the averages(5), the needed equidistribution properties on nilmanifolds present serious difficulties.…”