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

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.