Ergodic averages, taken along cubes whose sizes tend to infinity, converge in L^2

Analysis, Combinatorics 1 Minute

You need to know: Set {\mathbb Z}^k of vectors x=(x_1,\dots,x_k) with integer coordinates, addition in {\mathbb Z}^k, set {\mathbb N} of natural numbers, \limsup notation.

Background: The upper density d(A) of set A \subset {\mathbb N} is d(A)=\limsup\limits_{N\to \infty} \frac{1}{N}|A \cap \{1,2,\dots,N\}|. For any A \subset {\mathbb Z}^k and x\in {\mathbb Z}^k the translate A+x is \{y: \, y=a+x, \, a\in A \}. Let t(A) be the minimal number of translates of A needed to fully cover {\mathbb Z}^k. Set A is called syndetic if t(A)<\infty. Also, denote V_k \subset {\mathbb Z}^k the set of vectors x=(x_1,\dots,x_k) with all coordinates 0 or 1.

The Theorem: On 16th June 2002, Bernard Host and Bryna Kra submitted to the Annals of Mathematics a paper in which they proved that for any A \subset {\mathbb N} with d(A) > \delta > 0 and integer k \geq 1, the set of n=(n_1, n_2, \dots, n_k) \in {\mathbb Z}^k such that d\left(\bigcap\limits_{\epsilon\in V_k}\left(A + \sum\limits_{i=1}^k\epsilon_i n_i\right)\right) \geq \delta^{2^k} is syndetic.

Short context: The Theorem, as stated above, is a combinatorial reformulation of a deep theorem is the field of ergodic theory, which establishes L^2-convergence of so-called “ergodic averages taken along cubes whose sizes tend to infinity”. The details of this original formulation are too difficult to be presented here.

Links: The original paper is available here. See also Section 5.1 of this book for an accessible description of the Theorem.

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Published by Bogdan Grechuk

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