Some ergodic theorems involving Omega function and their applications

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 $$a,a+d,\dots ,a+kd,a+\Omega (d)\in A.$$

Correct an error in the proof of proposition 6.2

References in corpus (1)

• Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions