Polynomial averages converge to the product of integrals

By Nikos Frantzikinakis, Bryna Kra. 2004

69 citations recorded by the source corpus.

We answer a question posed by Vitaly Bergelson, showing that in a totally ergodic system, the average of a product of functions evaluated along polynomial times, with polynomials of pairwise differing degrees, converges inL2 to the product of the integrals. Such averages are characterized by nilsystems and so we reduce the problem to one of uniform distribution of polynomial sequences on nilmanifolds.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b13eacf20e2da2b29a7392b25e752d9dd03f63d0

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

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Citations in the corpus

Convergence of polynomial ergodic averages (2005, methodology): “We show that these factors are of the form of the factors Zk(X) introduced in [HK02] and we use properties proven in [ HK02 ] to describe them as inverse limits of nilsystems.”
Pointwise ergodic theorems for non-conventional bilinear polynomial averages (2020, background): “t;1by Holder’s inequality, and the case of general p 1;:::;p k then follows by a standard limiting argument using (1.3). There is a long history of prior partial results (e.g., [38], [75], [2], [5], [27], [32]) towards Theorem1.10, as well as generalizations to actions of other nilpotent groups than”
Multiple ergodic averages for three polynomials and applications (2006, unknown): “Here is a more detailed sketch of how this plan is executed to deal with the various parts of Theorem B: Part ( i ) deals with linearly independent polynomial families, a case that has already been worked out in [16] and [17] using a “reduction to aﬃne argument”.”
Ergodic averages of commuting transformations with distinct degree polynomial iterates (2009, methodology): “The proof in [15] is given for ergodic systems, but the announced result follows directly from this, since f⊥Krat(T, μ) implies that f⊥Krat(T, μx) = 0 for μ-almost every x ∈ X, where as usual, μ = ∫ μx dμ(x) is the ergodic decomposition of μ.”
The primes contain arbitrarily long polynomial progressions (2006, methodology): “1) can be established via the machinery of characteristic factors and Gowers–Host–Kra seminorms, by modifying the arguments in [9] and [21], and this does not require the axiom of choice; this already allows us to establish Theorem 1.”
Some open problems on multiple ergodic averages (2011, background): “, `n}) = ` − 1 ([31, 45]), and dmin(P) = 1 when P consists of at least two rationally independent polynomials ([22, 24]).”