Some ergodic theorems involving Omega function and their applications
| Autor: | Xiao, Rongzhong |
|---|---|
| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $$a,a+d,\ldots,a+kd,a+\Omega(d)\in A.$$ Comment: Correct an error in the proof of proposition 6.2 |
| Databáze: | arXiv |
| Externí odkaz: |
|