Algebraic approximations to linear combinations of S-units
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917087436341248 |
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| author | Nair, Parvathi S Kumar, Veekesh Rout, S. S. |
| author_facet | Nair, Parvathi S Kumar, Veekesh Rout, S. S. |
| contents | Let $Γ\subset \bar{\Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers, let $α_1,\ldots,α_m$ be non-zero algebraic numbers, and let $\varepsilon >0$ be fixed. In this paper, we prove that there exist only finitely many tuples $(u_1, \ldots, u_m, q, p)\in Γ^m\times\mathbb{Z}^2$ with $d = [\mathbb{Q}(u_1, \ldots, u_m):\mathbb{Q}]$ such that for any two tuples $(u_1,\ldots,u_m)$ and $(u'_1,\ldots,u'_m)$, we have $\frac{u_{i_1}}{u_{i_2}}\neq \frac{u'_{i_1}}{u'_{i_2}}$ for $1\leq i_1\neq i_2\leq m$ and it is stable under Galois conjugation over $\Q$, $\max\{|α_1 qu_1|, \ldots, |α_m qu_m|\}>1$, the tuple $(α_1qu_1, \ldots, α_mq u_m)$ is not pseudo-Pisot and \[0< \left|\sum_{i=1}^m α_iq u_i - p\right|<\frac{1}{\left(\prod_{i=1}^mH( u_i)\right)^{\varepsilon} |q|^{md+\varepsilon}},\] where $H(u_i)$ denotes the absolute Weil height. This result extends one of the main results of Corvaja-Zannier \cite{corv}. In addition, we prove a result similar to \cite[Theorem 1.4]{kul} in a more general setting. In our proofs, we exploit the subspace theorem based on the work of Corvaja-Zannier. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_02898 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebraic approximations to linear combinations of S-units Nair, Parvathi S Kumar, Veekesh Rout, S. S. Number Theory 11J68 (Primary ) 11J87, 11R06 (Secondary) Let $Γ\subset \bar{\Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers, let $α_1,\ldots,α_m$ be non-zero algebraic numbers, and let $\varepsilon >0$ be fixed. In this paper, we prove that there exist only finitely many tuples $(u_1, \ldots, u_m, q, p)\in Γ^m\times\mathbb{Z}^2$ with $d = [\mathbb{Q}(u_1, \ldots, u_m):\mathbb{Q}]$ such that for any two tuples $(u_1,\ldots,u_m)$ and $(u'_1,\ldots,u'_m)$, we have $\frac{u_{i_1}}{u_{i_2}}\neq \frac{u'_{i_1}}{u'_{i_2}}$ for $1\leq i_1\neq i_2\leq m$ and it is stable under Galois conjugation over $\Q$, $\max\{|α_1 qu_1|, \ldots, |α_m qu_m|\}>1$, the tuple $(α_1qu_1, \ldots, α_mq u_m)$ is not pseudo-Pisot and \[0< \left|\sum_{i=1}^m α_iq u_i - p\right|<\frac{1}{\left(\prod_{i=1}^mH( u_i)\right)^{\varepsilon} |q|^{md+\varepsilon}},\] where $H(u_i)$ denotes the absolute Weil height. This result extends one of the main results of Corvaja-Zannier \cite{corv}. In addition, we prove a result similar to \cite[Theorem 1.4]{kul} in a more general setting. In our proofs, we exploit the subspace theorem based on the work of Corvaja-Zannier. |
| title | Algebraic approximations to linear combinations of S-units |
| topic | Number Theory 11J68 (Primary ) 11J87, 11R06 (Secondary) |
| url | https://arxiv.org/abs/2506.02898 |