Some ergodic theorems involving Omega function and their applications
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909909801500672 |
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| author | Xiao, Rongzhong |
| author_facet | Xiao, Rongzhong |
| contents | In this paper, we build some ergodic theorems involving function $Ω$, where $Ω(n)$ 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 $$a,a+d,\ldots,a+kd,a+Ω(d)\in A.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03852 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some ergodic theorems involving Omega function and their applications Xiao, Rongzhong Dynamical Systems Combinatorics Number Theory In this paper, we build some ergodic theorems involving function $Ω$, where $Ω(n)$ 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 $$a,a+d,\ldots,a+kd,a+Ω(d)\in A.$$ |
| title | Some ergodic theorems involving Omega function and their applications |
| topic | Dynamical Systems Combinatorics Number Theory |
| url | https://arxiv.org/abs/2412.03852 |