Pointwise Ergodic Averages Along the Omega Function in Number Fields

TL;DR

This paper investigates the convergence properties of ergodic averages along the Omega function in number fields, demonstrating failure in some cases but establishing convergence in others, with implications for number theory and dynamical systems.

Contribution

It shows the failure of pointwise convergence of certain averages along the Omega function in number fields and proves convergence in uniquely ergodic systems using number-theoretic methods.

Findings

Averages along Omega function do not converge pointwise in general ergodic systems.

Convergence is established for averages on ideals in number fields within uniquely ergodic systems.

The results have implications for number-theoretic problems and dynamical systems analysis.

Abstract

We show the failure of the pointwise convergence of averages along the Omega function in a number field. As a consequence, we show, for instance, that the averages \[ \frac{1}{N^2}\sum_{1\leq m,n \leq N} f(T^{\Omega(m^2+n^2)}x)\] do not converge pointwise in ergodic systems, addressing a question posed by Le, Moreira, Sun, and the second author. On the other hand, using number-theoretic methods, we establish the pointwise convergence of averages along the $\Omega$ function defined on the ideals of a number field in uniquely ergodic systems. Using this dynamical framework, we also derive several natural number-theoretic consequences of independent interest.