Henry Towsner

Convergence of diagonal ergodic averages

Tao has recently proved that if T 1 ,…, T l are commuting, invertible, measure-preserving transformations on a dynamical system, then for any L ∞ functions f 1 ,…, f l , the average (1/ N )∑ n =0 N −1 ∏ i ≤ l f i ∘ T i n converges in the L 2 norm. Tao’s proof is unusual in that it translates the problem into a more complicated statement about the combinatorics of finite spaces by using the Furstenberg correspondence ‘backwards’. In this paper, we give an ergodic proof of this theorem, essentially a translation of Tao’s argument to the ergodic setting. In order to do this, we develop two new variations on the usual Furstenberg correspondence, both of which take recurrence-type statements in one dynamical system and give equivalent statements in a different dynamical system with desirable properties.

RELATIVELY EXCHANGEABLE STRUCTURES

We study random relational structures that are \emph{relatively exchangeable}---that is, whose distributions are invariant under the automorphisms of a reference structure

Epsilon Substitution for Transfinite Induction

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Ordinal Analysis by Transformations

. When

A realizability interpretation for classical analysis

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Local stability of ergodic averages

has {\em trivial definable closure}, every relatively exchangeable structure satisfies a general Aldous--Hoover-type representation. If

Functional interpretation and inductive definitions

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Hindman's Theorem: An Ultrafilter Argument in Second Order Arithmetic

satisfies the stronger properties of {\em ultrahomogeneity} and {\em

A combinatorial proof of the Dense Hindman's Theorem

n

Metastability in the Furstenberg-Zimmer tower

-disjoint amalgamation property} (