$\ell^p(\mathbb{Z})$ -- 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.