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Pointwise Ergodic Averages Along the Omega Function in Number Fields

Diego Céspedes, Sebastián Donoso

TL;DR

This work analyzes ergodic averages taken along the Omega and omega functions in number fields. It shows that pointwise convergence can fail for averages over ideals in non-atomic systems, while providing a positive convergence theory in uniquely ergodic settings and establishing norm convergence via a transference principle. The authors extend Erdős–Kac and Sathé–Selberg phenomena to number fields, adapt Loyd’s sweeping-out technique to ideals and to Gaussian integers, and derive number-theoretic corollaries such as density results and equidistribution statements for fractional parts of multiples of these arithmetic functions. Overall, the paper builds a robust dynamical-arithmetic framework linking ergodic averages to prime-ideal statistics, yielding both convergence results and concrete arithmetic consequences.

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 \[ \frac{1}{N^2}\sum_{1\leq m,n \leq N} f(T^{Ω(m^2+n^2)}x)\] 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 $Ω$ 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.

Pointwise Ergodic Averages Along the Omega Function in Number Fields

TL;DR

This work analyzes ergodic averages taken along the Omega and omega functions in number fields. It shows that pointwise convergence can fail for averages over ideals in non-atomic systems, while providing a positive convergence theory in uniquely ergodic settings and establishing norm convergence via a transference principle. The authors extend Erdős–Kac and Sathé–Selberg phenomena to number fields, adapt Loyd’s sweeping-out technique to ideals and to Gaussian integers, and derive number-theoretic corollaries such as density results and equidistribution statements for fractional parts of multiples of these arithmetic functions. Overall, the paper builds a robust dynamical-arithmetic framework linking ergodic averages to prime-ideal statistics, yielding both convergence results and concrete arithmetic consequences.

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 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 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.
Paper Structure (13 sections, 25 theorems, 85 equations)

This paper contains 13 sections, 25 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Extensions to other number fields
  3. Preliminaries
  4. Notation
  5. Asymptotic behavior of $\omega_{\mathbb{K}}(\cdot)$ and $\Omega_\mathbb{K}(\cdot)$
  6. Measure-preserving systems
  7. Failure of pointwise convergence
  8. Failure of pointwise convergence for number fields
  9. Failure of pointwise convergence along $\Omega(m^2+n^2)$.
  10. Convergence in uniquely ergodic systems and applications
  11. Convergence in the topological setting
  12. Norm convergence
  13. Applications to number theory

Key Result

Theorem 1

Let $(X, T)$ be a uniquely ergodic system and let $\mu$ be its unique invariant measure. Then, for any $f\in C(X)$ and any $x \in X$

Theorems & Definitions (34)

  • Theorem 1: Bergelson_Richter_dynamical_pnt:2022
  • Theorem 2: Loyd_dynamical_omega:2023
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8: Erdős-Kac theorem for number fields
  • Theorem 9: Hardy-Ramanujan for number fields
  • Lemma 1
  • ...and 24 more