Large sets of integers and hierarchy of mixing properties of measure preserving systems

Vitaly Bergelson, and Tomasz Downarowicz. "Large sets of integers and hierarchy of mixing properties of measure preserving systems." Colloquium Mathematicae 110.1 (2008): 117-150. <http://eudml.org/doc/283924>.

@article{VitalyBergelson2008,
abstract = {We consider a hierarchy of notions of largeness for subsets of ℤ (such as thick sets, syndetic sets, IP-sets, etc., as well as some new classes) and study them in conjunction with recurrence in topological dynamics and ergodic theory. We use topological dynamics and topological algebra in βℤ to establish connections between various notions of largeness and apply those results to the study of the sets $R^\{ε\}_\{A,B\} = \{n ∈ ℤ: μ(A ∩ TⁿB) > μ(A)μ(B) - ε\}$ of times of “fat intersection”. Among other things we show that the sets $R^\{ε\}_\{A,B\}$ allow one to distinguish between various notions of mixing and introduce an interesting class of weakly but not mildly mixing systems. Some of our results on fat intersections are established in a more general context of unitary ℤ-actions.},
author = {Vitaly Bergelson, Tomasz Downarowicz},
journal = {Colloquium Mathematicae},
keywords = {weak mixing; mild mixing; fat intersections; IP-sets; idempotents; central sets; upper Banach density; recurrence},
language = {eng},
number = {1},
pages = {117-150},
title = {Large sets of integers and hierarchy of mixing properties of measure preserving systems},
url = {http://eudml.org/doc/283924},
volume = {110},
year = {2008},
}

TY - JOUR
AU - Vitaly Bergelson
AU - Tomasz Downarowicz
TI - Large sets of integers and hierarchy of mixing properties of measure preserving systems
JO - Colloquium Mathematicae
PY - 2008
VL - 110
IS - 1
SP - 117
EP - 150
AB - We consider a hierarchy of notions of largeness for subsets of ℤ (such as thick sets, syndetic sets, IP-sets, etc., as well as some new classes) and study them in conjunction with recurrence in topological dynamics and ergodic theory. We use topological dynamics and topological algebra in βℤ to establish connections between various notions of largeness and apply those results to the study of the sets $R^{ε}_{A,B} = {n ∈ ℤ: μ(A ∩ TⁿB) > μ(A)μ(B) - ε}$ of times of “fat intersection”. Among other things we show that the sets $R^{ε}_{A,B}$ allow one to distinguish between various notions of mixing and introduce an interesting class of weakly but not mildly mixing systems. Some of our results on fat intersections are established in a more general context of unitary ℤ-actions.
LA - eng
KW - weak mixing; mild mixing; fat intersections; IP-sets; idempotents; central sets; upper Banach density; recurrence
UR - http://eudml.org/doc/283924
ER -