Algebraic approximations to linear combinations of powers: An extension of results by Mahler and Corvaja--Zannier
For every complex number x, let \Vert x\Vert _{\mathbb{Z}}:=\min \{\vert x-m\vert:\ m\in \mathbb{Z}\}. Let K be a number field, let k\in \mathbb{N}, and let \alpha _1,\ldots ,\alpha _k be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of \theta \in (0,1)...
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| Veröffentlicht in: | Transactions of the American Mathematical Society 2019-03, Vol.371 (6), p.3787-3804 |
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| container_title | Transactions of the American Mathematical Society |
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| creator | KULKARNI, AVINASH MAVRAKI, NIKI MYRTO NGUYEN, KHOA D. |
| description | For every complex number x, let \Vert x\Vert _{\mathbb{Z}}:=\min \{\vert x-m\vert:\ m\in \mathbb{Z}\}. Let K be a number field, let k\in \mathbb{N}, and let \alpha _1,\ldots ,\alpha _k be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of \theta \in (0,1) such that there are infinitely many tuples (n,q_1,\ldots ,q_k) satisfying \Vert q_1\alpha _1^n+\cdots +q_k\alpha _k^n\Vert _{\mathbb{Z}} |
| doi_str_mv | 10.1090/tran/7316 |
| format | Article |
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Let K be a number field, let k\in \mathbb{N}, and let \alpha _1,\ldots ,\alpha _k be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of \theta \in (0,1) such that there are infinitely many tuples (n,q_1,\ldots ,q_k) satisfying \Vert q_1\alpha _1^n+\cdots +q_k\alpha _k^n\Vert _{\mathbb{Z}}<\theta ^n where n\in \mathbb{N} and q_1,\ldots ,q_k\in K^* have small logarithmic height compared to n. In the special case when q_1,\ldots ,q_k have the form q_i=qc_i for fixed c_1,\ldots ,c_k, our work yields results on algebraic approximations of c_1\alpha _1^n+\cdots +c_k\alpha _k^n of the form \frac {m}{q} with m\in \mathbb{Z} and q\in K^* (where q has small logarithmic height compared to n). Various results on linear recurrence sequences also follow as an immediate consequence. The case where k=1 and q_1 is essentially a rational integer was obtained by Corvaja and Zannier and settled a long-standing question of Mahler. 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Let K be a number field, let k\in \mathbb{N}, and let \alpha _1,\ldots ,\alpha _k be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of \theta \in (0,1) such that there are infinitely many tuples (n,q_1,\ldots ,q_k) satisfying \Vert q_1\alpha _1^n+\cdots +q_k\alpha _k^n\Vert _{\mathbb{Z}}<\theta ^n where n\in \mathbb{N} and q_1,\ldots ,q_k\in K^* have small logarithmic height compared to n. In the special case when q_1,\ldots ,q_k have the form q_i=qc_i for fixed c_1,\ldots ,c_k, our work yields results on algebraic approximations of c_1\alpha _1^n+\cdots +c_k\alpha _k^n of the form \frac {m}{q} with m\in \mathbb{Z} and q\in K^* (where q has small logarithmic height compared to n). Various results on linear recurrence sequences also follow as an immediate consequence. The case where k=1 and q_1 is essentially a rational integer was obtained by Corvaja and Zannier and settled a long-standing question of Mahler. The use of the Subspace Theorem based on work of Corvaja-Zannier, together with several modifications, plays an important role in the proof of our results.</abstract><pub>American Mathematical Society</pub><doi>10.1090/tran/7316</doi><tpages>18</tpages><oa>free_for_read</oa></addata></record> |
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| title | Algebraic approximations to linear combinations of powers: An extension of results by Mahler and Corvaja--Zannier |
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