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The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters

  • Mustafa Akcoglu (a1), Alexandra Bellow (a2), Roger L. Jones (a3), Viktor Losert (a4), Karin Reinhold-Larsson (a5) and Máté Wierdl (a2)
  • DOI: http://dx.doi.org/10.1017/S0143385700008798
  • Published online: 01 September 2008
Abstract
Abstract

In this paper we establish conditions on a sequence of operators which imply divergence. In fact, we give conditions which imply that we can find a set B of measure as close to zero as we like, but such that the operators applied to the characteristic function of this set have a lim sup equal to 1 and a lim inf equal to 0 a.e. (strong sweeping out). The results include the fact that ergodic averages along lacunary sequences, certain convolution powers, and the Riemann sums considered by Rudin are all strong sweeping out. One of the criteria for strong sweeping out involves a condition on the Fourier transform of the sequence of measures, which is often easily checked. The second criterion for strong sweeping out involves showing that a sequence of numbers satisfies a property similar to the conclusion of Kronecker's lemma on sequences linearly independent over the rationals.

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[ABdJJ]M. Akcoglu , A. Bellow , A. del Junco and R. Jones . Divergence of averages obtained by sampling a flow. Proc. A.M.S. 118 (1993), 499505

[AdJ]M. Akcoglu and A. del Junco . Convergence of averages of point trsnsformations. Proc. A.M.S. 49 (1975), 265266

[BBB]V. Bergelson , J. Bourgain and M. Boshernitzan . Some results on non-linear recurrence. J. d'Anal. Math. 62 (1994), 2946

[Bo1]J. Bourgain . Almost sure convergence and bounded entropy. Israel J. Math. 63 (1988), 7997

[Bo2]J. Bourgain , Problems of almost everywhere convergence related to harmonic analysis and number theory. Israel J. of Math., 71 (1990), 97127

[Bo3]J. Bourgain . On the maximal ergodic theorem for certain subsets of the integers. Israel J. Math. 61 (1988), 3972

[dJR]A. del Junco and J. Rosenblatt . Counter examples in ergodic theory and number theory. Math. Ann. 245 (1979), 185197

[F]H. Furstenberg . Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, 1981.

[K1]U. Krengel . Ergodic Theorems, de Gruyter, Berlin-New York, 1985.

[K2]U. Krengel . On the individual ergodic theorem for subsequences. Ann. Math. Stat. 42 (1971), 10911095

[L]D. Lind . Locally compact measure preserving flows. Adv. in Math. 15 (1975), 175193

[R1]J. Rosenblatt . Ergodic group actions. Arch. Math. 47 (1986), 263269

[R]G. C. Rota . An ‘Alternierende Verfahren’ for general positive operators. Bull. A.M.S. 68 (1962), 95102

[Ru]W. Rudin . An arithmetic property of Riemann sums. P. A. M. S. 15 (1964), 321324

[St]N. Starr . Operator limit theorems. Trans. A.M.S. 121 (1966), 90115

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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