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Pointwise ergodic averages along the Omega function in number fields
[Submitted on 22 Jan 2026 (v1), last revised 25 Jun 2026 (this v · 2026-06-26 · via math updates on arXiv.org

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Abstract:We study a general criterion for guaranteeing that an ergodic average exhibits the strong sweeping-out property. This result implies, in particular, the failure of pointwise convergence of these averages. Our result applies to averages along the Omega function in number fields, generalizing a result of Loyd. We also show that the averages \[\frac{1}{N^2}\sum_{1\leq m,n \leq N}f(T^{\Omega(m^2+n^2)}x)\] exhibit the strong sweeping-out property, which answers affirmatively 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.

Submission history

From: Sebastián Donoso [view email]
[v1] Thu, 22 Jan 2026 17:31:56 UTC (18 KB)
[v2] Thu, 25 Jun 2026 17:30:05 UTC (21 KB)

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