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D sets and IP rich sets in ℤ

Randall McCutcheonJee Zhou — 2016

Fundamenta Mathematicae

We give combinatorial characterizations of IP rich sets (IP sets that remain IP upon removal of any set of zero upper Banach density) and D sets (members of idempotent ultrafilters, all of whose members have positive upper Banach density) in ℤ. We then show that the family of IP rich sets strictly contains the family of D sets.

Aspects of uniformity in recurrence

Vitaly BergelsonBernard HostRandall McCutcheonFraniçois Parreau — 2000

Colloquium Mathematicae

We analyze and cite applications of various, loosely related notions of uniformity inherent to the phenomenon of (multiple) recurrence in ergodic theory. An assortment of results are obtained, among them sharpenings of two theorems due to Bourgain. The first of these, which in the original guarantees existence of sets x,x+h, x + h 2 in subsets E of positive measure in the unit interval, with lower bounds on h depending only on m(E), is expanded to the case of arbitrary finite polynomial configurations...

An Alpern tower independent of a given partition

James T. CampbellJared T. CollinsSteven KalikowRaena KingRandall McCutcheon — 2015

Colloquium Mathematicae

Given a measure-preserving transformation T of a probability space (X,ℬ,μ) and a finite measurable partition ℙ of X, we show how to construct an Alpern tower of any height whose base is independent of the partition ℙ. That is, given N ∈ ℕ, there exists a Rokhlin tower of height N, with base B and error set E, such that B is independent of ℙ, and TE ⊂ B.

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