Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some ergodic theorems involving Omega function and their applications

Rongzhong Xiao

TL;DR

This work develops ergodic theorems involving the Omega function $\Omega(n)$, establishing weighted mean convergence results for products of measure-preserving transformations with $S^{\Omega(n)}$ and deriving substantial corollaries for rotations on tori, Liouville-type functions, and unipotent affine actions. The approach hinges on pronilfactors, nilsequences, and an orthogonality criterion to control $L^{2}$-limits and enable applications to polynomial Szemerédi-type theorems. It further translates ergodic results into combinatorial consequences via Furstenberg’s correspondence, yielding additive patterns in sets of positive upper Banach density, such as $a,a+d,\dots,a+kd,a+\Omega(d)$. The paper concludes with open questions about extending Omega-patterns to higher-dimensional configurations and recurrence-times, inviting further exploration of $\Omega(n)$ in additive combinatorics and ergodic theory.

Abstract

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 $$a,a+d,\ldots,a+kd,a+Ω(d)\in A.$$

Some ergodic theorems involving Omega function and their applications

TL;DR

This work develops ergodic theorems involving the Omega function , establishing weighted mean convergence results for products of measure-preserving transformations with and deriving substantial corollaries for rotations on tori, Liouville-type functions, and unipotent affine actions. The approach hinges on pronilfactors, nilsequences, and an orthogonality criterion to control -limits and enable applications to polynomial Szemerédi-type theorems. It further translates ergodic results into combinatorial consequences via Furstenberg’s correspondence, yielding additive patterns in sets of positive upper Banach density, such as . The paper concludes with open questions about extending Omega-patterns to higher-dimensional configurations and recurrence-times, inviting further exploration of in additive combinatorics and ergodic theory.

Abstract

In this paper, we build some ergodic theorems involving function , where denotes the number of prime factors of a natural number counted with multiplicities. As a combinatorial application, it is shown that for any and every with positive upper Banach density, there are such that

Paper Structure

This paper contains 14 sections, 20 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Isomorphism and factors
  4. Conditional expectation and ergodic decomposition
  5. Nilsequences
  6. Pronilfactors
  7. An othogonality criterion
  8. Proof Theorem \ref{['TA']}
  9. Applications of Theorem \ref{['TA']}
  10. Proofs of Theorem \ref{['TB']} and Theorem \ref{['TC']}
  11. Proofs of Proposition \ref{['PA']} - \ref{['PC']}
  12. Some questions
  13. On Proposition \ref{['PA']}
  14. On recurrence times

Key Result

Theorem 1.1

$($BR22$)$ Let $(X,T)$ be a uniquely ergodic topological dynamical system with unique $T$-invariant Borel probability measure $\mu$. Then for any $f\in C(X),x\in X$,

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Remark 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 17 more