November 9, 2025Open Access

A 2-adic Density Lemma in the Odd Trajectory of the Collatz Map

Key Points

Relative density for odd numbers is shown to be 1/2^(r-1), revealing unique patterns in arithmetic progressions.
Key finding includes the association of numerical families, leading all odd numbers to a common global attractor.
Observational analysis connects the Collatz dynamics with 4-adic families, demonstrating convergence behavior.
Insights from this framework highlight the significance of the density concept, potentially impacting probabilistic models.

Abstract

This work analyzes the odd–even dynamics of the classical Collatz map from a 2-adic viewpoint. It proves that odd numbers with ν₂ (3n + 1) = r form arithmetic progressions of relative density 1/2^r–1, explaining the increasing gaps observed in Collatz trajectories. Using this 2-adic information, all odd numbers sharing the same last even term are grouped into 4-adic families, showing that every family converges to 𝔽₂ = 1, 5, 21, 85, …. The cycle 4 → 2 → 1 emerges as a unique global attractor. The framework also connects explicitly with the Structure Theorem for (d, g, h) -maps by Kontorovich and Sinai (2006), where the decreasing 2-adic density plays the role of the negative drift in their probabilistic model.

A 2-adic Density Lemma in the Odd Trajectory of the Collatz Map

Key Points

Abstract

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