Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-15T14:18:45.719Z Has data issue: false hasContentIssue false
  • English
  • Français

On polynomials in primes and J. Bourgain's circle method approach to ergodic theorems

Published online by Cambridge University Press:  19 September 2008

R. Nair
R. Nair
Affiliation:
Department of Pure Mathematics, University of Liverpool, PO Box 147, Liverpool L69 3BX, UK
Get accessRights & Permissions [Opens in a new window]

Extract

In this paper we prove the following theorem.

Theorem 1. For a measure-preserving system (X, β, μ, T) and a positive integer k, if f ∈ L2(X, β, μ), the averages

,

converge μ almost everywhere. Here p runs over the rational primes and πN denotes their number in [1, N].

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 11 , Issue 3 , September 1991, pp. 485 - 499
Copyright
Copyright © Cambridge University Press 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Apostol, T. M.. Introduction to Analytic Number theory, Undergraduate Texts in Mathematics, Springer-Verlag: 1976.Google Scholar
[2]Bellow, A.. On bad universal sequences in ergodic theory (II). Proc. Sherbrooke Workshop on Measure Theory, Springer Lecture Notes in Mathematics 1033 (1984), 74–78.Google Scholar
[3]Bourgain, J.. On the maximal ergodic theorem for certain subsets of the integers. Isr. J. Math. 61 (1988), 39–72.Google Scholar
[4]Bourgain, J.. Pointwise ergodic theorems for arithmetic sets. Publ. Math. IHES 69 (1989), 5–45.CrossRefGoogle Scholar
[5]Bourgain, J.. An approach to pointwise ergodic theorems. G.A.F.A. Seminar 1987. Springer-Verlag Lecture Notes in Mathematics 1317 (1988), 20–223.Google Scholar
[6]Calderon, A. P.. Ergodic theory and translation invariant operators. Proc. Nat. Acad. Sci. 59 (1968), 349–353.CrossRefGoogle ScholarPubMed
[7]Davenport, H.. Multiplicative number theory, Springer-Verlag Graduate Texts in Mathematics 74 (1980).Google Scholar
[8]Hua, L. K.. Additive theory of prime numbers. Amer. Math. Soc. Trans. 13 (1965).Google Scholar
[9]Rhin, G.. Répartition modulo 1 de f(p n) quand f est une series entière. Springer-Verlag Lecture Notes in Mathematics 475 (1975), 176–244.Google Scholar
[10]Vinogradov, I. M.. Selected Works, Springer-Verlag: 1985.Google Scholar
[11]Weil, A.. On some exponential sums. Proc. Nat. Acad. Sci. Washington 34–5 (1948), 204–207.Google Scholar
[12]Weyl, H.. Über die Gleichvertilung von Zahlen mod. Eins. Math. Ann. 77 (1916), 313–352.Google Scholar
[13]Wierdl, M.. Pointwise ergodic theorem along the prime numbers. Isr. J. Math. 64 (3) (1988), 315–336.Google Scholar