The mean ergodic theorem is equivalent to the assertion that for every function K and every epsilon, there is an n with the property that the ergodic averages A_m f are stable to within epsilon on the interval [n,K(n)]. We show that even though it is not generally possible to compute a bound on the rate of convergence of a sequence of ergodic averages, one can give explicit bounds on n in terms of K and || f || / epsilon. This tells us how far one has to search to find an n so that the ergodic averages are "locally stable" on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general proof-theoretic methods falling under the heading of "proof mining."Comment: Minor errors corrected. To appear in Transactions of the AM

In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from RT, Ramsey's Theorem for Pairs. These principles tend to fall outside of the "big five" systems of reverse mathematics and a complicated picture of subsystems below RT has emerged. In this paper, we answer two open questions concerning these subsystems, specifically that ADS is not equivalent to CAC and that EM is not equivalent to RT.

We examine the correspondence between the various notions of quasirandomness for k‐uniform hypergraphs and σ‐algebras related to measurable hypergraphs. This gives a uniform formulation of most of the notions of quasirandomness for dense hypergraphs which have been studied, with each notion of quasirandomness corresponding to a σ‐algebra defined by a collection of subsets of [1,k]. We associate each notion of quasirandomness ℐ with a collection of hypergraphs, the ℐ‐adapted hypergraphs, so that G is quasirandom exactly when it contains roughly the correct number of copies of each ℐ‐adapted hypergraph. We then identify, for each ℐ, a particular ℐ‐adapted hypergraph Mk[ℐ] with the property that if G contains roughly the correct number of copies of Mk[ℐ] then G is quasirandom in the sense of ℐ. This generalizes recent results of Kohayakawa, Nagle, Rödl, and Schacht; Conlon, Hàn, Person, and Schacht; and Lenz and Mubayi giving this result for some particular notions of quasirandomness. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 114–139, 2017

Tao has recently proved that ifT1,…,Tlare commuting, invertible, measure-preserving transformations on a dynamical system, then for anyL∞functionsf1,…,fl, the average (1/N)∑n=0N−1∏i≤lfi∘Tinconverges in theL2norm. 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.

Abstract. We present a logical framework for formalizing connections between finitary combinatorics and measure theory or ergodic theory that have appeared in various places throughout the literature. We develop the basic syntax and semantics of this logic and give applications, showing that the method can express the classic Furstenberg correspondence and to short proofs of the Szemerédi Regularity Lemma and the hypergraph removal lemma. We also derive some connections between the model-theoretic notion of stability and the Gowers uniformity norms from combinatorics.