Pointwise ergodic theorems for some thin subsets of primes

TL;DR

This paper proves pointwise ergodic theorems for certain prime subsets using $ ext{l}^p$ boundedness of variation operators, advancing understanding of ergodic behavior along thin prime sets.

Contribution

It introduces new methods to establish pointwise ergodic theorems along specific thin subsets of primes by analyzing variation operators.

Findings

Established $ ext{l}^p$ boundedness of $r$-variations for prime subset operators.

Proved pointwise ergodic theorems for these prime subsets.

Extended ergodic theory to new classes of prime-related sequences.

Abstract

We establish pointwise ergodic theorems for operators of Radon type along subsets of prime numbers of the form $\big\{\{ \varphi_1(n)\} < \psi(n)\big\}$. We achieve this by proving $\ell^p(\mathbb{Z})$ boundedness of $r$-variations, where $p > 1$ and $r > 2$.