Some ergodic theorems involving Omega function and their applications

TL;DR

This paper develops new ergodic theorems involving the Omega function, which counts prime factors, and applies these results to combinatorial problems related to density and arithmetic progressions.

Contribution

It introduces ergodic theorems that incorporate the Omega function and demonstrates their application to combinatorial density problems.

Findings

Established ergodic theorems involving the Omega function.

Proved existence of arithmetic progressions with prime factor conditions in dense sets.

Connected ergodic theory with combinatorial number theory.

Abstract

In this paper, we build some ergodic theorems involving function $\Omega$, where $\Omega(n)$ denotes the number of prime factors of a natural number $n$ counted with multiplicities. As a combinatorial application, it is shown that for any $k\in \mathbb{N}$ and every $A\subset \mathbb{N}$ with positive upper Banach density, there are $a,d\in \mathbb{N}$ such that $$a,a+d,\ldots,a+kd,a+\Omega(d)\in A.$$