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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

  • Mariusz MirekEmail author
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.

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

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

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