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arXiv · 2412.03852

Some ergodic theorems involving Omega function and their applications

Rongzhong Xiao
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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 $$a,a+d,\ldots,a+kd,a+\Omega(d)\in A.$$

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Rongzhong Xiao. 2024-12-05. Some ergodic theorems involving Omega function and their applications. https://doi.org/10.1017/etds.2025.10254

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