Aspects of uniformity in recurrence

Volume 84 / 2000

Vitaly Bergelson, Bernard Host, Randall McCutcheon, Franiçois Parreau Colloquium Mathematicum 84 (2000), 549-576 DOI: 10.4064/cm-84/85-2-549-576

Abstract

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.

Authors

  • Vitaly Bergelson
  • Bernard Host
  • Randall McCutcheon
  • Franiçois Parreau

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