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Algebraically unrelated sequences. (English) Zbl 1048.11060

From the text: The paper deals with the so-called algebraically unrelated sequences defined as follows: The sequences \((a_{i,n})\in \mathbb R^\mathbb N_+ (i=1,\dots ,k)\) are called algebraically unrelated if, for any \((c_n)\in \mathbb N^\mathbb N\), the numbers \(\sum_{n=1}^\infty (a_{i,n} c_n)^{-1}\) \((i=1,\dots ,k)\) are algebraically independent. The main result runs as follows.
Let \((a_{i,n}), (b_{i,n}) \in \mathbb N^\mathbb N\) \((i= 1,\dots , k)\), and assume \((a_{1,n})\) to be non-decreasing such that, as \(n\to \infty\), \(a_{i,n} b_{j,n} = o (a_{j,n} b_{i,n})\) for any \((i,j)\) with \(1\leq j<i\leq k\), and \(\lim\sup n^{-1}\log\log a_{1,n} = \infty\). Assume further that, with fixed \(\varepsilon , \varepsilon_1,\varepsilon_2, \varepsilon_3\in R_+\) satisfying \((1- \varepsilon_1) \varepsilon_2 (1+ \varepsilon ) >1\) and \( \varepsilon_2 < 1 < \varepsilon_3 \), then for any large \(n\) and for any \(i=1,\dots ,k\) the following inequalities hold:
\(n^{1+\varepsilon}<a_{1,n}; b_{i,n}< a_{1,n}^{\varepsilon_1}\);
\(a_{1,n}^{\varepsilon_2} < a_{i,n} < a_{1,n}^{\varepsilon_3}\).
Then the sequences \((a_{i,n} / b_{i,n})\in\mathbb Q_+^\mathbb N\) \((i=1,\dots ,k)\) are algebraically unrelated.
Some applications for infinite series of rational numbers are included.

MSC:

11J85 Algebraic independence; Gel’fond’s method
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References:

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