The sequence of perfect squares is L1-universally bad

Analysis 2 Minutes

You need to know: Probability space X with measure \mu and set of measurable subsets {\cal X}, the notion of “almost all” x\in X, notation T^n(x) for n-fold composiition T(T(\dots T(x))\dots) of map T: X \to X, notation T^{-1}(A):=\{x \in X\,|\, T(x) \in A\} for the preimage of A \subset X, integration on X, notation L^p(X) for set of functions f:X \to {\mathbb R} such that \int_X|f(x)|^pd\mu < \infty, notation \{n_k\}_{k=1}^\infty for sequence n_1, n_2, \dots, n_k, \dots.

Background: An ergodic dynamical system (X,T) is a probability space X, together with map T: X \to X such that (i) \mu (T^{-1}(A)) = \mu(A) for all A \in {\cal X}, and (ii) for any A \in {\cal X} with T^{-1}(A)=A, either \mu(A)=0 or \mu(A)=1. A sequence \{n_k\}_{k=1}^\infty is called L1-universally bad if for all ergodic dynamical systems (X,T) there is some f \in L^1(X) and A \in {\cal X} with \mu(A)>0, such that the limit \lim\limits_{N\to\infty}\frac{1}{N}\sum\limits_{k=1}^N f(T^{n_k}(x)) fails to exist for all x\in A.

The Theorem: On 5th April 2005, Zoltán Buczolich and Daniel Mauldin submitted to arxiv and The Annals of Mathematics a paper in which they proved that the sequence \{k^2\}_{k=1}^\infty is L1-universally bad.

Short context: Famous Birkhoff’s Ergodic Theorem states that, for any ergodic dynamical system (X,T) and any f \in L^1(X) the limit \lim\limits_{N\to\infty}\frac{1}{N}\sum\limits_{k=1}^N f(T^k(x)) exists for almost all x\in X (in fact, this limit is equal to \int_X f(x) d\mu). An important research direction is to understand for which sequences \{n_k\}_{k=1}^\infty the corresponding limit \lim\limits_{N\to\infty}\frac{1}{N}\sum\limits_{k=1}^N f(T^{n_k}(x)) exists for almost all x\in X. In 1988, Bourgain proved this for sequence \{k^2\}_{k=1}^\infty, provided that f \in L^p(X) for some p>1, and asked if the same is true for all f \in L^1(X). The Theorem answers this question negatively.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 10.6 of this book for an accessible description of the Theorem.

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Published by Bogdan Grechuk

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