Pointwise ergodic averages along the Omega function in number fields

Technical Summary: Pointwise Ergodic Averages Along the Omega Function in Number Fields

Problem Statement
This paper investigates the convergence properties of ergodic averages along arithmetic sequences defined by the Omega function ($\Omega$) and the distinct prime factor function ($\omega$) within the context of number fields. The central problem addresses two contrasting behaviors in dynamical systems:

1. Failure of Pointwise Convergence: In general ergodic systems (specifically non-atomic, invertible measure-preserving systems), do averages along these arithmetic sequences fail to converge pointwise? Specifically, do they exhibit the "strong sweeping-out property," where the limsup of the average of a characteristic function is 1 and the liminf is 0 for almost every point?
2. Convergence in Uniquely Ergodic Systems: Do these averages converge pointwise to the space average when the underlying dynamical system is uniquely ergodic?

The authors extend previous results by Bergelson and Richter (for $\mathbb{Q}$) and Donoso et al. (for Gaussian integers) to arbitrary number fields and more complex averaging schemes, such as those involving sums of squares ($m^2+n^2$).

Methodology
The paper employs a synthesis of ergodic theory and analytic number theory. The methodology is divided into two primary strands:

1. Establishing Strong Sweeping-Out (Failure of Convergence):

   • The authors utilize a criterion introduced by del Junco and Rosenblatt, which relies on verifying a maximal inequality for the averaging operators.
   • To verify this inequality without direct computation on arbitrary systems, they employ a transfer principle (Proposition 2.11) that reduces the problem to checking the behavior of the averaging scheme on a finite interval of integers.
   • Crucially, the authors link the strong sweeping-out property to the Erdős–Kac theorem. They prove that if an averaging scheme satisfies an Erdős–Kac type theorem (i.e., the distribution of the arithmetic function, normalized by $\log_2 N$, converges to a standard Gaussian), then the scheme exhibits the strong sweeping-out property.
   • They apply number-theoretic results (Hardy–Ramanujan, Sathé–Selberg, and specific extensions by Liu and Wu) to demonstrate that $\Omega_K$ and $\omega_K$ on ideals of number fields, as well as $\Omega(m^2+n^2)$, satisfy these distributional conditions.

2. Establishing Pointwise Convergence:

   • For uniquely ergodic systems, the authors adopt an approach involving asymptotic shift-invariance.
   • They prove that for bounded sequences, the average of $f(T^{c(a)+1}x)$ is asymptotically equivalent to the average of $f(T^{c(a)}x)$. This relies on the statistical properties of $\Omega_K$ and $\omega_K$ (specifically the Hardy–Ramanujan theorem and the Sathé–Selberg asymptotic formulas) to show that the weight distribution of the functions shifts negligibly as $N \to \infty$.
   • This shift-invariance, combined with the unique ergodicity of the system, implies the convergence of the ergodic averages to the integral of the function with respect to the invariant measure.
   • Norm convergence is derived using spectral theory (Bochner–Herglotz theorem) and the fact that the averages converge to zero for irrational rotations.

Key Contributions and Results

• General Criterion for Strong Sweeping-Out (Theorem 3.5): The paper establishes that any averaging scheme satisfying an Erdős–Kac type theorem exhibits the strong sweeping-out property. This generalizes previous results by Loyd and Mondal.
• Extension to Arbitrary Number Fields (Theorem 1.4): The authors prove that for any number field $K$, the averages along $\Omega_K$ (total prime factors) and $\omega_K$ (distinct prime factors) of ideals satisfy the strong sweeping-out property in non-atomic ergodic systems. This generalizes Loyd's result from $\mathbb{Q}$ to all number fields.
• Resolution of Question 1 (Theorem 3.10): The paper answers affirmatively a question posed by Le, Moreira, Sun, and the second author regarding the averages $\frac{1}{N^2} \sum_{1 \le m,n \le N} f(T^{\Omega(m^2+n^2)}x)$. The authors prove these averages also exhibit the strong sweeping-out property by showing that $\Omega(m^2+n^2)$ satisfies an Erdős–Kac type theorem (Lemmas 3.8 and 3.9).
• Pointwise Convergence in Uniquely Ergodic Systems (Theorem 1.5): The authors prove that for any uniquely ergodic system $(X, T)$ and continuous function $f$, the averages along $\Omega_K$ and $\omega_K$ converge pointwise to $\int f d\mu$. This recovers the Bergelson–Richter theorem for $\mathbb{Q}$ and extends it to Gaussian integers and general number fields.
• Norm Convergence (Theorem 4.6): The paper establishes $L^2$ norm convergence for these averages in general measure-preserving systems.
• Number-Theoretic Corollaries: Using the dynamical convergence results, the authors derive equidistribution results for $\Omega_K$ and $\omega_K$ modulo $m$ (generalizing Pillai–Selberg) and for fractional parts $\{\alpha \Omega_K(a)\}$ (generalizing Erdős–Delange).

Significance and Claims
The paper claims to provide a unified framework for understanding the divergence and convergence of ergodic averages along arithmetic functions in number fields.

• Regarding Divergence: The authors assert that the strong sweeping-out property is a robust phenomenon for these averages, contingent only on the arithmetic function satisfying an Erdős–Kac distribution. This resolves the open question regarding the specific case of $\Omega(m^2+n^2)$ and generalizes the scope of known divergence results to all number fields.
• Regarding Convergence: The paper demonstrates that while pointwise convergence fails in general ergodic systems, it is restored in uniquely ergodic systems. The authors note that their proof for Theorem 1.5 relies on a strengthening of Landau's prime ideal theorem and does not provide a new proof of that theorem itself.
• Limitations and Open Problems: The authors modestly note that their methods are insufficient to prove convergence for other types of ergodic averages, such as those involving $\Omega(N(a))$ (where the argument is the norm itself rather than the ideal). They suggest that a "dynamical statement" concerning multiplicative actions defined by prime ideals would be required to resolve these cases, noting this remains an open problem.

In summary, the work bridges the gap between the statistical distribution of prime factors in number fields and the ergodic behavior of dynamical systems, confirming that the "strong sweeping-out" behavior is the generic outcome for these averages in non-atomic systems, while pointwise convergence is preserved in the uniquely ergodic setting.