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Polynomial averages converge to the product of integrals

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Abstract

We answer a question posed by Vitaly Bergelson, showing that in a totally ergodic system, the average of a product of functions evaluated along polynomial times, with polynomials of pairwise differing degrees, converges inL 2 to the product of the integrals. Such averages are characterized by nilsystems and so we reduce the problem to one of uniform distribution of polynomial sequences on nilmanifolds.

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Authors and Affiliations

  1. Department of Mathematics, McAllister Building, The Pennsylvania State University, 16802, University Park, PA, USA

    Nikos Frantzikinakis & Bryna Kra

Authors
  1. Nikos Frantzikinakis
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  2. Bryna Kra
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Corresponding author

Correspondence to Nikos Frantzikinakis.

Additional information

Dedicated to Hillel Furstenberg upon his retirement

The second author was partially supported by NSF grant DMS-0244994.

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Frantzikinakis, N., Kra, B. Polynomial averages converge to the product of integrals. Isr. J. Math. 148, 267–276 (2005). https://doi.org/10.1007/BF02775439

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  • DOI: https://doi.org/10.1007/BF02775439

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