$\left(p,q\right)$-adic Analysis and the Collatz Conjecture
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| Formato: | Preprint |
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2024
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| _version_ | 1866915047643545600 |
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| author | Siegel, Maxwell Charles |
| author_facet | Siegel, Maxwell Charles |
| contents | What use can there be for a function from the $p$-adic numbers to the $q$-adic numbers, where $p$ and $q$ are distinct primes? The traditional answer, courtesy of the half-century old theory of non-archimedean functional analysis: not much. It turns out this judgment was premature. '$\left(p,q\right)$-adic analysis' of this sort appears to be naturally suited for studying the infamous Collatz map and similar arithmetical dynamical systems. Given such a map $H:\mathbb{Z}\rightarrow\mathbb{Z}$, one can construct a function $χ_{H}:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q}$ for an appropriate choice of distinct primes $p,q$ with the property that $x\in\mathbb{Z}\backslash\left\{ 0\right\} $ is a periodic point of $H$ if and only if there is a $p$-adic integer $\mathfrak{z}\in\left(\mathbb{Q}\cap\mathbb{Z}_{p}\right)\backslash\left\{ 0,1,2,\ldots\right\} $ so that $χ_{H}\left(\mathfrak{z}\right)=x$. By generalizing Monna-Springer integration theory and establishing a $\left(p,q\right)$-adic analogue of the Wiener Tauberian Theorem, one can show that the question 'is $x\in\mathbb{Z}\backslash\left\{ 0\right\} $ a periodic point of $H$?' is essentially equivalent to 'is the span of the translates of the Fourier transform of $χ_{H}\left(\mathfrak{z}\right)-x$ dense in an appropriate non-archimedean function space?' This presents an exciting new frontier in Collatz research, and these methods can be used to study Collatz-type dynamical systems on the lattice $\mathbb{Z}^{d}$ for any $d\geq1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02902 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\left(p,q\right)$-adic Analysis and the Collatz Conjecture Siegel, Maxwell Charles General Mathematics 11S80 (Primary), 37P99 (Primary), 46S10 (Secondary) What use can there be for a function from the $p$-adic numbers to the $q$-adic numbers, where $p$ and $q$ are distinct primes? The traditional answer, courtesy of the half-century old theory of non-archimedean functional analysis: not much. It turns out this judgment was premature. '$\left(p,q\right)$-adic analysis' of this sort appears to be naturally suited for studying the infamous Collatz map and similar arithmetical dynamical systems. Given such a map $H:\mathbb{Z}\rightarrow\mathbb{Z}$, one can construct a function $χ_{H}:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q}$ for an appropriate choice of distinct primes $p,q$ with the property that $x\in\mathbb{Z}\backslash\left\{ 0\right\} $ is a periodic point of $H$ if and only if there is a $p$-adic integer $\mathfrak{z}\in\left(\mathbb{Q}\cap\mathbb{Z}_{p}\right)\backslash\left\{ 0,1,2,\ldots\right\} $ so that $χ_{H}\left(\mathfrak{z}\right)=x$. By generalizing Monna-Springer integration theory and establishing a $\left(p,q\right)$-adic analogue of the Wiener Tauberian Theorem, one can show that the question 'is $x\in\mathbb{Z}\backslash\left\{ 0\right\} $ a periodic point of $H$?' is essentially equivalent to 'is the span of the translates of the Fourier transform of $χ_{H}\left(\mathfrak{z}\right)-x$ dense in an appropriate non-archimedean function space?' This presents an exciting new frontier in Collatz research, and these methods can be used to study Collatz-type dynamical systems on the lattice $\mathbb{Z}^{d}$ for any $d\geq1$. |
| title | $\left(p,q\right)$-adic Analysis and the Collatz Conjecture |
| topic | General Mathematics 11S80 (Primary), 37P99 (Primary), 46S10 (Secondary) |
| url | https://arxiv.org/abs/2412.02902 |