Skip to main content
Log in

On polynomials in primes, ergodic averages and monothetic groups

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript


Let G denote a compact monothetic group, and let \(\rho (x) = \alpha _k x^k + \ldots + \alpha _1 x + \alpha _0\), where \(\alpha _0, \ldots , \alpha _k\) are elements of G one of which is a generator of G. Let \((p_n)_{n\ge 1}\) denote the sequence of rational prime numbers. Suppose \(f \in L^{p}(G)\) for \(p> 1\). It is known that if

$$\begin{aligned} A_{N}f(x):= {1 \over N} \sum _{n=1}^{N} f(x + \rho (p_n)) \quad (N=1,2, \ldots ), \end{aligned}$$

then the limit \(\lim _{n\rightarrow \infty } A_Nf(x)\) exists for almost all x with respect Haar measure. We show that if G is connected then the limit is \(\int _{G} f d\lambda \). In the case where G is the a-adic integers, which is a totally disconnected group, the limit is described in terms of Fourier multipliers which are generalizations of Gauss sums.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Apostol, T.M.: Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, p. xii+338. Springer, New York-Heidelberg (1976)

    Book  Google Scholar 

  2. Asmar, N., Nair, R.: Certain averages on the a-adic numbers. Proc. Am. Math Soc. vol. 114(1), 21–28 (1992)

    MathSciNet  Google Scholar 

  3. Bourgain, J.: An approach to pointwise ergodic theorems. In: Geometric Aspects of Functional Analysis (1986/87). Lecture Notes in Math., vol. 1317, pp. 204–223. Springer, Berlin (1988)

  4. Bourgain, J.: On the maximal ergodic theorem for certain subsets of the integers. Isr. J. Math. 61, 39–72 (1988)

    Article  MathSciNet  Google Scholar 

  5. Bourgain, J.: Pointwise ergodic theorems for arithmetic sets, I. H. É. S. Publ. Math. 69, 5–45 (1989)

    Article  MathSciNet  Google Scholar 

  6. Buczolich, Z., Mauldin, R.D.: Divergent square averages. Ann. Math. (2) 71(3), 1479–1530 (2010)

    Article  MathSciNet  Google Scholar 

  7. Davenport, H.: Multiplicative Number Theory, Graduate Texts in Mathematics, vol. 74, 2nd edn. Springer (1980)

  8. Dikranjan, D., Bruno, A.G.: Compact groups with a dense free abelian subgroup. Rend. Istit. Mat. Univer. Trieste 45, 137–150 (2013)

    MathSciNet  Google Scholar 

  9. Falcone, G., Plaumann, P., Strambach, K.: Monothetic algebraic groups. J. Aust. Math. Soc. 82(3), 315–324 (2007)

    Article  MathSciNet  Google Scholar 

  10. Graham, C.C., McGehee, O.C.: Essays in Commutative Harmonic Analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 238, pp. xxi+464. Springer, New York-Berlin (1979)

  11. Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, vol. I. Springer, UK (1980)

    Google Scholar 

  12. Koksma, J.F., Salem, R.: Uniform distribution and Lebesgue integration. Acta Sci. Math. Szeged 12, (1950). Leopoldo Fejér et Frederico Riesz LXX annos natis dedicatus, Pars B, 87–96 = Math. Centrum Amsterdam. Rapport ZW 1949–004, 9pp (1949)

  13. Kuipers, H., Neiderreiter, H.: Uniform Distribution of Sequences. Wiley (1973)

  14. Marstrand, J.: On Khinchin’s conjecture about strong uniform distribution. Proc. Lond. Math. Soc. 21, 514–556 (1970)

    MathSciNet  Google Scholar 

  15. Nair, R.: On polynomial ergodic averages and square functions. In: Number Theory and Polynomials. London Math. Soc. Lecture Note Ser., vol. 352, pp. 241–254. Cambridge University Press, Cambridge (2008)

  16. Nair, R.: On polynomials in primes and J. Bourgain’s circle method approach to ergodic theorems. Ergod. Theory Dyn. Syst. 11, 485–499 (1989)

    Article  MathSciNet  Google Scholar 

  17. Nair, R.: On polynomials in primes and J. Bourgain’s circle method approach to ergodic theorems II. Stud. Math. 105(3), 207–233 (1993)

    Article  MathSciNet  Google Scholar 

  18. Nair, R.: On some arithmetic properties of \(L^p\) summable functions. Q. J. Math. Oxford 47(2), 101–105 (1996)

    Article  Google Scholar 

  19. Nair, R.: On asymptotic distribution on the a-adic integers. Proc. Indian. Acad. Sci. 107, 363–376 (1997)

    Article  MathSciNet  Google Scholar 

  20. Nienhuys, J.W.: Some examples of monothetic groups. Fund. Math. 88(2), 163–171 (1975)

    Article  MathSciNet  Google Scholar 

  21. Rhin, G.: Sur la répartition modulo 1 des suites \(f(p)\). Acta Arith. XXII I, 217–248 (1973)

    Article  MathSciNet  Google Scholar 

  22. Trojan, B.: Variational estimates for discrete operators modeled on multi-dimensional polynomial subsets of primes. Math. Ann. 374(3–4), 1597–1656 (2019)

    Article  MathSciNet  Google Scholar 

  23. van Dantzig, D.: Über topologisch homogene Kontinua. Fund. Math. 15, 102–125 (1930)

    Article  Google Scholar 

  24. Weyl, H.: Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77, 313–352 (1916)

    Article  MathSciNet  Google Scholar 

  25. Wierdl, M.: Pointwise ergodic theorem along the prime numbers. Israel J. Math. 64(3), 315–336 (1989)

    Article  MathSciNet  Google Scholar 

Download references


The authors thank Buket Eren Gökmen for comments that much improved the readability of the manuscript. Radhakrishnan Nair thanks Laboratoire de Mathématique de l’Université Savoie Mont Blanc for its hospitality and financial support while this paper was being written We also thank the referee for very detailed comments which materially improved the paper.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Jean-Louis Verger-Gaugry.

Additional information

Communicated by Karlheinz Gröchenig.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hančl, J., Nair, R. & Verger-Gaugry, JL. On polynomials in primes, ergodic averages and monothetic groups. Monatsh Math 204, 47–62 (2024).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification