An Explicit Near-Conjugacy Between the Collatz Map and a Circle Rotation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909984275562496 |
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| author | Asli, Barmak Honarvar Shakibaei |
| author_facet | Asli, Barmak Honarvar Shakibaei |
| contents | We introduce an explicit logarithmic transformation $T(x) = \{\log_6(x + 1/5)\}$ under which the Collatz map becomes a rigid circle rotation by the irrational angle \(α= \log_6 3\), perturbed by a uniformly bounded error term. We prove that for all positive integers \(x\), $T(C(x)) = T(x) + α+ \varepsilon(x) \pmod{1}$, where \(|\varepsilon(x)| \le 0.2749\) and \(\varepsilon(x) = O(1/x)\) as \(x \to \infty\). We derive the transformation from an exact functional equation linking the even and odd branches of the Collatz map, explain the arithmetic origin of the parameters \(6\) and \(1/5\), and analyse the structure of the resulting error term. Extensive numerical computations up to \(10^{12}\) confirm the sharpness of the bounds and show that cumulative errors remain uniformly bounded along all tested trajectories. While this near-conjugacy does not by itself resolve the Collatz conjecture, it provides a concrete and quantitative dynamical framework that clarifies the geometric structure underlying the Collatz iteration and may be useful in further analytical or experimental investigations of Collatz-type systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_04289 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Explicit Near-Conjugacy Between the Collatz Map and a Circle Rotation Asli, Barmak Honarvar Shakibaei General Mathematics We introduce an explicit logarithmic transformation $T(x) = \{\log_6(x + 1/5)\}$ under which the Collatz map becomes a rigid circle rotation by the irrational angle \(α= \log_6 3\), perturbed by a uniformly bounded error term. We prove that for all positive integers \(x\), $T(C(x)) = T(x) + α+ \varepsilon(x) \pmod{1}$, where \(|\varepsilon(x)| \le 0.2749\) and \(\varepsilon(x) = O(1/x)\) as \(x \to \infty\). We derive the transformation from an exact functional equation linking the even and odd branches of the Collatz map, explain the arithmetic origin of the parameters \(6\) and \(1/5\), and analyse the structure of the resulting error term. Extensive numerical computations up to \(10^{12}\) confirm the sharpness of the bounds and show that cumulative errors remain uniformly bounded along all tested trajectories. While this near-conjugacy does not by itself resolve the Collatz conjecture, it provides a concrete and quantitative dynamical framework that clarifies the geometric structure underlying the Collatz iteration and may be useful in further analytical or experimental investigations of Collatz-type systems. |
| title | An Explicit Near-Conjugacy Between the Collatz Map and a Circle Rotation |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2601.04289 |