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1
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D sets and IP rich sets in ℤ

100%
McCutcheon R., Zhou J.
Fundamenta Mathematicae
|
2016
|
tom 233
|
nr 1
71-82
EN
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.
2
Content available remote

Simultaneous Diophantine approximation and VIP systems

81%
Bergelson V., Knutson I., McCutcheon R.
Acta Arithmetica
|
2005
|
tom 116
|
nr 1
13-23
3
Content available remote

Aspects of uniformity in recurrence

81%
Bergelson V., Host B., McCutcheon R., Parreau F.
Colloquium Mathematicum
|
2000
|
tom 84/85
|
nr 2
549-576
EN
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 in subsets of positive measure in cubes of $ℝ^{n}$. The second is a direct computation of a lower bound, uniform in a and b and depending only on ∫f, for ∫f(x)f(x+at)f(x+bt)dxdt, where 0≤f≤1 is a function on the 1-torus. Our methodology parallels that of Bourgain, who originally considered the case a=1, b=2.
4
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An Alpern tower independent of a given partition

71%
Campbell J., Collins J., Kalikow S., King R., McCutcheon R.
Colloquium Mathematicum
|
2015
|
tom 141
|
nr 1
119-124
EN
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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