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

TL;DR

This paper proves pointwise ergodic theorems along sparse sets of primes, including Piatetski-Shapiro primes, and applies these methods to solve the ternary Goldbach problem for certain thin prime sets.

Contribution

It introduces the first pointwise ergodic theorems along zero-density prime subsets and extends techniques to address the ternary Goldbach problem in this context.

Findings

Established pointwise ergodic theorems for thin prime sets

Applied methods to solve the ternary Goldbach problem for specific prime subsets

Demonstrated robustness of techniques for sparse prime sequences

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.