Directional $p$-Adic Littlewood Conjecture for Algebraic Vectors
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arXiv
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| Format: | Preprint |
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2025
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| author | Yifrach, Yuval |
| author_facet | Yifrach, Yuval |
| contents | For every vector $\overline α\in \RR^n$ and for every rational approximation $(\overline p,q)\in \RR^n\times\RR$ we can associate the displacement vector $qα-\overline p$. We focus on algebraic vectors, namely $\overline α=(α_1,\dots,α_n)$ such that $1, α_1, \dots, α_n$ span a rank $n$ number field. For these vectors, we investigate the size of their displacements as well as the distribution of their directions. We give a new proof to the result of Bugeaud in \cite{YannPAdic} saying that algebraic vectors $\overline α$ satisfy the $p$-adic Littlewood Conjecture. Namely, we prove that \begin{equation}
\liminf_{k \to \infty} \left( k \abs{k}_p \right)^{1/n} \| k (α_1, \dots, α_n) \|_\infty = 0. \end{equation} Our new proof lets us classify all limiting distributions, with a special weighting, of the sequence of directions of the defects in the $\varepsilon$-approximations of $(α_1, \dots, α_n)$. Each such limiting measure is expressed as the pushforward of an algebraic measure on $X_n$ to the sphere. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_04430 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Directional $p$-Adic Littlewood Conjecture for Algebraic Vectors Yifrach, Yuval Dynamical Systems Number Theory 11J04, 37A05 For every vector $\overline α\in \RR^n$ and for every rational approximation $(\overline p,q)\in \RR^n\times\RR$ we can associate the displacement vector $qα-\overline p$. We focus on algebraic vectors, namely $\overline α=(α_1,\dots,α_n)$ such that $1, α_1, \dots, α_n$ span a rank $n$ number field. For these vectors, we investigate the size of their displacements as well as the distribution of their directions. We give a new proof to the result of Bugeaud in \cite{YannPAdic} saying that algebraic vectors $\overline α$ satisfy the $p$-adic Littlewood Conjecture. Namely, we prove that \begin{equation} \liminf_{k \to \infty} \left( k \abs{k}_p \right)^{1/n} \| k (α_1, \dots, α_n) \|_\infty = 0. \end{equation} Our new proof lets us classify all limiting distributions, with a special weighting, of the sequence of directions of the defects in the $\varepsilon$-approximations of $(α_1, \dots, α_n)$. Each such limiting measure is expressed as the pushforward of an algebraic measure on $X_n$ to the sphere. |
| title | Directional $p$-Adic Littlewood Conjecture for Algebraic Vectors |
| topic | Dynamical Systems Number Theory 11J04, 37A05 |
| url | https://arxiv.org/abs/2501.04430 |