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Aspects of uniformity in recurrence

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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.
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  • Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, OH 43210, U.S.A.
  • Équipe d'Analyse et de Mathématiques Appliquées, Université de Marne-la-Vallée, 5 Boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallée, Cedex 2, France
  • Department of Mathematics, University of Maryland, College Park, MD 20742, U.S.A.
  • LAGA, UMR CNRS 7539, Université de Paris 13, 93430 Villetaneuse, France
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