Aspects of uniformity in recurrence

Vitaly Bergelson; Bernard Host; Randall McCutcheon; Franiçois Parreau

ArticleOriginal scientific text

Title

Aspects of uniformity in recurrence

Authors ¹, ², ³, ⁴

Affiliations

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

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.

[B] V. Bergelson, Sets of recurrence of $ℤ^{m}$ -actions and properties of sets of differences in $ℤ^{m}$ , J. London Math. Soc. (2) 31 (1985), 295-304.
[BBB] V. Bergelson, M. Boshernitzan, and J. Bourgain, Some results on non-linear recurrence, J. Anal. Math. 62 (1994), 29-46.
[BH] V. Bergelson and I. Håland, Sets of recurrence and generalized polynomials, in: Convergence in Ergodic Theory and Probability, June 1993, Ohio State University Math. Research Institute Publications, de Gruyter, 1996, 91-110.
[BL] V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden's and Szemerédi's theorem, J. Amer. Math. Soc. 9 (1996), 725-753.
[Bo1] J. Bourgain, A Szemerédi type theorem for sets of positive density in $ℝ^{k}$ , Israel J. Math. 54 (1986), 307-316.
[Bo2] J. Bourgain, A non-linear version of Roth's theorem for sets of positive density in the real line, J. Anal. Math. 50 (1988), 169-181.
[Fo] A. Forrest, Recurrence in dynamical systems: a combinatorial approach, Ph.D. Thesis, Ohio State University, 1990.
[F] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, ibid. 31 (1977), 204-256.
[FK] H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, ibid. 34 (1978), 275-291.
[FKO] H. Furstenberg, Y. Katznelson and D. Ornstein, The ergodic theoretical proof of Szemerédi's theorem, Bull. Amer. Math. Soc. 7 (1982), 527-552.
[G1] W. T. Gowers, A new proof of Szemerédi's theorem for arithmetic progressions of length four, Geom. Funct. Anal. 8 (1998), 529-551.
[G2] W. T. Gowers, Fourier analysis and Szemerédi's theorem, in: Proc. Internat. Congress Math., Vol. I, Doc. Math. (Berlin, 1998), 617-629.
[G3] W. T. Gowers, A new proof of Szemerédi's theorem, Geom. Funct. Anal., to appear.
[H-B] D. R. Heath-Brown, Integer sets containing no arithmetic progressions, J. London Math. Soc. (2) 35 (1987), 385-394.
[La] Y. Lacroix, private communication.
[L] D. Lind, Locally compact measure preserving flows, Adv. Math. 15 (1975), 175-193.
[R] K. Roth, Sur quelques ensembles d'entiers, C. R. Acad. Sci. Paris 234 (1952), 388-390.
[S1] E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 199-245.
[S2] E. Szemerédi, Integer sets containing no arithmetic progressions, Acta Math. Hungar. 56 (1990), 155-158.

Title

Aspects of uniformity in recurrence

Affiliations

Abstract

Bibliography