Aspects of uniformity in recurrence

Vitaly Bergelson; Bernard Host; Randall McCutcheon; Franiçois Parreau

Colloquium Mathematicae (2000)

  • Volume: 84/85, Issue: 2, page 549-576
  • ISSN: 0010-1354

Abstract

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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.

How to cite

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Bergelson, Vitaly, et al. "Aspects of uniformity in recurrence." Colloquium Mathematicae 84/85.2 (2000): 549-576. <http://eudml.org/doc/210832>.

@article{Bergelson2000,
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.},
author = {Bergelson, Vitaly, Host, Bernard, McCutcheon, Randall, Parreau, Franiçois},
journal = {Colloquium Mathematicae},
keywords = {multiple recurrence; multidimensional Szemerédi theorem; uniformity; ergodic theory},
language = {eng},
number = {2},
pages = {549-576},
title = {Aspects of uniformity in recurrence},
url = {http://eudml.org/doc/210832},
volume = {84/85},
year = {2000},
}

TY - JOUR
AU - Bergelson, Vitaly
AU - Host, Bernard
AU - McCutcheon, Randall
AU - Parreau, Franiçois
TI - Aspects of uniformity in recurrence
JO - Colloquium Mathematicae
PY - 2000
VL - 84/85
IS - 2
SP - 549
EP - 576
AB - 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.
LA - eng
KW - multiple recurrence; multidimensional Szemerédi theorem; uniformity; ergodic theory
UR - http://eudml.org/doc/210832
ER -

References

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  13. [G3] W. T. Gowers, A new proof of Szemerédi's theorem, Geom. Funct. Anal., to appear. Zbl1028.11005
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  15. [La] Y. Lacroix, private communication. 
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