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Van der Corput sets in $ℤ^{d}$

100%

Bergelson V. , Lesigne E.

nr 1

1-49

In this partly expository paper we study van der Corput sets in $ℤ^{d}$, with a focus on connections with harmonic analysis and recurrence properties of measure preserving dynamical systems. We prove multidimensional versions of some classical results obtained for d = 1 by Kamae and M. Mendès France and by Ruzsa, establish new characterizations, introduce and discuss some modifications of van der Corput sets which correspond to various notions of recurrence, provide numerous examples and formulate some natural open questions.

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

100%

Bergelson V. , Downarowicz T.

nr 1

117-150

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.

Aspects of uniformity in recurrence

80%

Bergelson V. , Host B. , McCutcheon R. , Parreau F.

nr 2

549-576

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.

Simultaneous Diophantine approximation and VIP systems

80%

Bergelson V. , Knutson I. , McCutcheon R.

nr 1

13-23