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
Full Text:

References:

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