Polynomial averages converge to the product of integrals

Nikos Frantzikinakis; Bryna Kra

doi:10.1007/BF02775439

Polynomial averages converge to the product of integrals

Nikos Frantzikinakis^*, Bryna Kra

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

34 Scopus citations

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 in L² 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.

Original language	English (US)
Pages (from-to)	267-276
Number of pages	10
Journal	Israel Journal of Mathematics
Volume	148
DOIs	https://doi.org/10.1007/BF02775439
State	Published - 2005

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Mathematics(all)

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10.1007/BF02775439

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title = "Polynomial averages converge to the product of integrals",

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 in L2 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.",

author = "Nikos Frantzikinakis and Bryna Kra",

note = "Funding Information: * The second author was partially supported by NSF grant DMS-0244994. Received October 2, 2003 and in revised form November 11, 2003",

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journal = "Israel Journal of Mathematics",

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AU - Frantzikinakis, Nikos

AU - Kra, Bryna

N1 - Funding Information: * The second author was partially supported by NSF grant DMS-0244994. Received October 2, 2003 and in revised form November 11, 2003

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Y1 - 2005

N2 - 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 in L2 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.

AB - 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 in L2 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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JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

ER -

Polynomial averages converge to the product of integrals

Abstract

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