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Some ergodic theorems involving Omega function and their applications

Part of: Ergodic theory

Published online by Cambridge University Press:  17 November 2025

RONGZHONG XIAO
RONGZHONG XIAO*
Affiliation:
University of Science and Technology of China School of Mathematical Sciences , China
*
e-mail: xiaorz@mail.ustc.edu.cn
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Abstract

In this paper, we build some ergodic theorems involving the 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 $a,a+d,\ldots, a+kd,a+\Omega(d)\in A.$

Keywords

Ergodic theoremsmultiple recurrenceOmega functionpolynomial Szemerédi theorem

MSC classification

Primary: 37A30: Ergodic theorems, spectral theory, Markov operators

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 20
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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