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

Volume 110 / 2008

Vitaly Bergelson, Tomasz Downarowicz Colloquium Mathematicum 110 (2008), 117-150 MSC: 37A25, 37B20. DOI: 10.4064/cm110-1-4


We consider a hierarchy of notions of largeness for subsets of ${\mathbb Z}$ (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 $\beta{\mathbb Z}$ to establish connections between various notions of largeness and apply those results to the study of the sets $R^\varepsilon_{A,B} = \{n\in{\mathbb Z}: \mu(A\cap T^nB)>\mu(A)\mu(B) - \varepsilon\}$ of times of “fat intersection”. Among other things we show that the sets $R^\varepsilon_{A,B}$ 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 ${\mathbb Z}$-actions.


  • Vitaly BergelsonDepartment of Mathematics
    The Ohio State University
    100 Math Tower
    231 West 18th Avenue
    Columbus, OH 43210-1174, U.S.A.
  • Tomasz DownarowiczInstitute of Mathematics
    Technical University
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland

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