Almost-Boundedness of Collatz-Type Maps on Number Rings via Archimedean Methods

Abstract

Let p be an integer ≥2, not necessarily prime, and let K be a number field. In 2019, Tao showed that almost every positive integer (in the sense of logarithmic density) had an almost bounded trajectory under the Collatz map. In this paper, we generalize this result to a large class of Collatz-type maps (p-Hydras) acting on the ring of integers 𝒪_K of a given number field K of dimension d over ℚ. Given an ideal Λ⊆𝒪_K of index p, a p-Hydra map H:𝒪_K→𝒪_K is a collection of affine-linear maps { H_j:K→ K} _{j∈𝒪_K/Λ} of the form z↦ r_jz+c_j for constants c_j∈ K and invertible ℚ-linear maps r_j:K→ K so that the map 1−r₀ is invertible and so that H(z) is defined to be H_{[z]_Λ}(z), where [z]_Λ is the projection of z mod Λ. We show that the direct analogue of Tao's almost-boundedness result holds provided, in addition to several minor conditions, that the Dimension Constraint (p−1)²ρ_H^d<1 is satisfied, where ρ_H=∏_j=0^p−1‖ r_j‖ _Mink is the product of the operator norms of the r_js with respect to the Minkowski embedding of K. Unlike Tao's original argument, which uses a $3$-adic Fourier decay result for the Syracuse Random Variables, our approach employs analysis of an archimedean flavor, and does not require any Fourier decay estimates for our generalized Syracuse Random Variables. Our result includes Tao's original result as a special case.

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Cite

BibTeX RIS

BibTeX

@misc{siegel2026almostboundedness,
  title = {{Almost-Boundedness of Collatz-Type Maps on Number Rings via Archimedean Methods}},
  author = {Maxwell Siegel and Rory O'Dwyer},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33018},
  eprint = {2609.33018},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33018},
  note = {CC BY-SA 4.0},
}

RIS

TY  - UNPB
TI  - Almost-Boundedness of Collatz-Type Maps on Number Rings via Archimedean Methods
AU  - Maxwell Siegel
AU  - Rory O'Dwyer
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.33018
UR  - https://arxiv.org/abs/2609.33018
AB  - Let p be an integer ≥2, not necessarily prime, and let K be a number field. In 2019, Tao showed that almost every positive integer (in the sense of logarithmic density) had an almost bounded trajectory under the Collatz map. In this paper, we generalize this result to a large class of Collatz-type maps (p-Hydras) acting on the ring of integers 𝒪_K of a given number field K of dimension d over ℚ. Given an ideal Λ⊆𝒪_K of index p, a p-Hydra map H:𝒪_K→𝒪_K is a collection of affine-linear maps { H_j:K→ K} _j∈𝒪_K/Λ of the form z↦ r_jz+c_j for constants c_j∈ K and invertible ℚ-linear maps r_j:K→ K so that the map 1−r_0 is invertible and so that H(z) is defined to be H_[z]_Λ(z), where [z]_Λ is the projection of z mod Λ. We show that the direct analogue of Tao's almost-boundedness result holds provided, in addition to several minor conditions, that the Dimension Constraint (p−1)^2ρ_H^d<1 is satisfied, where ρ_H=∏_j=0^p−1‖ r_j‖ _Mink is the product of the operator norms of the r_js with respect to the Minkowski embedding of K. Unlike Tao's original argument, which uses a $3$-adic Fourier decay result for the Syracuse Random Variables, our approach employs analysis of an archimedean flavor, and does not require any Fourier decay estimates for our generalized Syracuse Random Variables. Our result includes Tao's original result as a special case.
ER  -

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