Mathematische Zeitschrift

, Volume 279, Issue 1–2, pp 27–59 | Cite as

\(\ell ^p({\mathbb {Z}})\)—Boundedness of discrete maximal functions along thin subsets of primes and pointwise ergodic theorems

Article

Abstract

We establish the first pointwise ergodic theorems along thin sets of prime numbers; a set with zero density with respect to the primes. For instance we will be able to achieve this with the Piatetski–Shapiro primes. Our methods will be robust enough to solve the ternary Goldbach problem for some thin sets of primes.

Notes

Acknowledgments

The author is greatly indebted to Christoph Thiele and Jim Wright for many helpful suggestions improving the exposition of this paper. The author is grateful to the referee for careful reading of the manuscript and useful remarks that led to the improvement of the presentation.

References

  1. 1.
    Balog, A., Friedlander, J.P.: A hybrid of theorems of Vinogradov and Piatetski-Shapiro. Pac. J. Math. 156, 45–62 (1992)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Boshernitzan, M., Kolesnik, G., Quas, A., Wierdl, M.: Ergodic averaging sequences. J. d’Analyse Math. 95(1), 63–103 (2005)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bourgain, J.: On the maximal ergodic theorem for certain subsets of the integers. Israel J. Math. 61, 39–72 (1988)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bourgain, J.: On the pointwise ergodic theorem on \(L^p\) for arithmetic sets. Israel J. Math. 61, 73–84 (1988)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bourgain, J.: Pointwise ergodic theorems for arithmetic sets, with an appendix by the author, H. Furstenberg, Y. Katznelson, and D. S. Ornstein. Inst. Hautes Etudes Sci. Publ. Math. 69, 5–45 (1989)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Buczolich, Z., Mauldin, R.D.: Divergent square averages. Ann. Math. 171(3), 1479–1530 (2010)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Graham, S.W., Kolesnik, G.: Van der Corputs Method of Exponential Sums. London Mathematical Society Lecture Note Series, 126. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar
  8. 8.
    Heath-Brown, D.R.: The Pjateckii-Sapiro prime number theorem. J. Number Theory 16, 242–266 (1983)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Helfgott, H.A.: Minor arcs for Goldbach’s problem. http://arxiv.org/abs/1205.5252
  10. 10.
    Helfgott, H.A.: Major arcs for Goldbach’s theorem. http://arxiv.org/abs/1305.2897
  11. 11.
    Helfgott, H.A.: The ternary Goldbach conjecture is true. http://arxiv.org/abs/1312.7748
  12. 12.
    Ionescu, A.D., Magyar, A., Stein, E.M., Wainger, S.: Discrete Radon transforms and applications to ergodic theory. Acta Math. 198, 231–298 (2007)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Ionescu, A.D., Wainger, S.: \(L^p\) boundedness of discrete singular Radon transforms. J. Am. Math. Soc. 19(2), 357–383 (2005)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Iwaniec, H., Kowalski, E.: Analytic Number Theory, vol. 53. Amer. Math. Soc. Colloquium Publications, Providence RI (2004)MATHGoogle Scholar
  15. 15.
    Kontorovich, A.V.: A Pseudo Twin Primes Theorem. In: Multiple Dirichlet Series, \(L\)-functions and Automorphic Forms. Progress in Math Series, vol.300, pp. 287–298. Birkhäuser, Boston (2012)Google Scholar
  16. 16.
    Kolesnik, G.: The distribution of primes in sequences of the form \(\lfloor n^c\rfloor \). Mat. Zametki 2, 117–128 (1967)MATHMathSciNetGoogle Scholar
  17. 17.
    Kolesnik, G.: Primes of theform \(\lfloor n^c\rfloor \). Pac. J. Math. 118, 437–447 (1985)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Kumchev, A.: On the Piatetski-Shapiro-Vinogradov theorem. Journal de Théorie des Nombres de Bordeaux 9(1), 11–23 (1997)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    LaVictoire, P.: Universally \(L^1\)-bad arithmetic sequences. J. Anal. Math. 113(1), 241–263 (2011)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Leitmann, D.: The distribution of prime numbers in sequences of the form \(\lfloor f(n)\rfloor \). Proc. Lond. Math. Soc. 35(3), 448–462 (1977)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Liu, H.Q., Rivat, J.: On the Piateski-Shapiro prime number theorem. Bull. Lond. Math. Soc. 24, 143–147 (1992)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Magyar, A., Stein, E.M., Wainger, S.: Discrete analogues in harmonic analysis: spherical averages. Ann. Math. 155, 189–208 (2002)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Mirek, M.: Roth’s Theorem in the Piatetski-Shapiro primes. Accepted for publication in Revista Matemática Iberoamericana. http://arxiv.org/abs/1305.0043 (2014)
  24. 24.
    Mirek, M.: Weak type \((1, 1)\) inequalities for discrete rough maximal functions. Accepted for publication in Journal d’Analyse Mathematique. http://arxiv.org/abs/1305.0575 (2014)
  25. 25.
    Mirek, M., Trojan, B.: Cotlar’s ergodic theorem along the set of prime numbers. Preprint. http://arxiv.org/abs/1311.7572 (2014)
  26. 26.
    Nathanson, M.B.: Additive Number Theory. The Classical Bases. Springer, Berlin (1996)CrossRefMATHGoogle Scholar
  27. 27.
    Nair, R.: On polynomials in primes and J. Bourgain’s circle method approach to ergodic theorems. Ergod. Theory Dyn. Syst. 11, 485–499 (1991)CrossRefMATHGoogle Scholar
  28. 28.
    Nair, R.: On polynomials in primes and J. Bourgain’s circle method approach to ergodic theorems II. Studia Mathematica 105(3), 207–233 (1993)MATHMathSciNetGoogle Scholar
  29. 29.
    Oberlin, D.M.: Two discrete fractional integrals. Math. Res. Lett. 8, 1–6 (2001)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Piatetski-Shapiro, I.: On the distribution of prime numbers in sequences of the form \(\lfloor f(n)\rfloor \). Math. Sbornik 33, 559–566 (1953)Google Scholar
  31. 31.
    Pierce, L.B.: Discrete analogues in harmonic analysis. PhD Thesis, Princeton University (2009)Google Scholar
  32. 32.
    Rivat, J., Sargos, P.: Nombres premiers de la forme \(\lfloor n^c\rfloor \). Can. J. Math. 53, 414–433 (2001)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Rosenblatt, J., Wierdl, M.: Pointwise ergodic theorems via harmonic analysis. Ergodic theory and its connections with harmonic analysis (Alexandria, 1993). 3–151, London Math. Soc. Lecture Note Ser., 205, Cambridge University Press, Cambridge (1995)Google Scholar
  34. 34.
    Stein, E.M., Wainger, S.: Discrete analogues of singular Radon transforms. Bull. Am. Math. Soc. 23, 537–544 (1990)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Stein, E.M., Wainger, S.: Discrete analogues in harmonic analysis I: \(\ell ^2\) estimates for singular Radon transforms. Am. J. Math. 121, 1291–1336 (1999)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Stein, E.M., Wainger, S.: Discrete analogues in harmonic analysis II: fractional integration. J. d’Analyse Math. 80, 335–355 (2000)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Stein, E.M., Wainger, S.: Two discrete fractional integral operators revisited. J. d’Analyse Math. 87, 451–479 (2002)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Urban, R., Zienkiewicz, J.: Weak type \((1,1)\) estimates for a class of discrete rough maximal functions. Math. Res. Lett. 14(2), 227–237 (2007)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    van der Corput, J.G.: Neue zahlentheoretische Abschatzungen II. Math. Zeit. 29, 397–426 (1929)CrossRefGoogle Scholar
  40. 40.
    Wainger, S.: Discrete analogues of singular and maximal Radon transforms. Doc. Math. Extra Volume ICM II pp. 743–753 (1998)Google Scholar
  41. 41.
    Wierdl, M.: Pointwise ergodic theorem along the prime numbers. Israel J. Math. 64(3), 315–336 (1988)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Mathematical InstituteUniversität BonnBonnGermany
  2. 2.Mathematical InstituteUniwersytet WrocławskiWrocławPoland

Personalised recommendations